Geometry and combinatorics of cube complexes

Geometry and combinatorics of cube
complexes
Mark F. Hagen
Doctor of Philosophy
Department of Mathematics and Statistics
McGill University
Montreal, Quebec
February 9, 2012
A thesis submitted to McGill University in partial fulfillment of the
requirements of the degree of Doctor of Philosophy.
c Mark F. Hagen, 2011.
⃝
ACKNOWLEDGEMENTS
The following people are hereby sincerely thanked. My advisor, Dani
Wise, has exhibited great patience, tolerance, humor, and expository gifts,
and made vigorous efforts to share his own ideas with me and to listen to
mine. It is a real pleasure to work with such an excellent mathematician and
human being. I am grateful to the illustrious mathematician and fisherman,
Victor Chepoi, for carefully reading a draft of the paper on which part of this
thesis is based, and for offering a great deal of valuable criticism, eventually
leading to a fruitful and enjoyable collaboration through which I was exposed
to a fresh viewpoint on this subject. I thank Michah Sageev for reading and
discussing [Hag11], for suggesting several interesting questions (only one of
which is really answered in this document), for telling me about the bottleneck
property, and for his hospitality during my visit to the Technion in September
2010. Jason Behrstock and Mikaël Pichot read the initial draft of this thesis
and offered several helpful comments. Piotr Przytycki read and astutely
criticized some of this work, as did Martin Bridson and two anonymous
referees, and I am grateful for their input. I thank Dominique Rabet for fixing
my French, below. I also thank McGill University and its Department of
Mathematics and Statistics for financial support throughout my time here.
My parents, Martha Foley and Steve Hagen have provided all kinds of love
and fascination for a subjective forever. The friendly and interesting graduate
students at McGill has been a source of great entertainment and support,
especially Hadi Bigdely, Sara Froehlich, Bahare Mirza, Atefeh Mohajeri,
Dominique Rabet, and Benjamin Smith. Will Bangs, Liam Capper-Starr,
Greta Hagen, Zoë Hagen, David Kurtz, Dave Morgan, Tim Nest, and other
friends are all deeply appreciated people. I am infinitely glad and grateful that
Janine Bachrachas has a large supply of psycho-spiritual spoons and is willing
to share.
ii
ABSTRACT
We study the geometry of median graphs and CAT(0) cube complexes by
introducing two combinatorial objects: the contact graph and the simplicial
boundary. The first of these encodes the intersections of hyperplane-carriers.
We prove that this graph is always quasi-isometric to a tree, and deduce that
groups acting properly and cocompactly on cube complexes are weakly hyperbolic relative to the collection of hyperplane-stabilizers. Using diagrammatic
techniques of Casson-Sageev-Wise, we study complete bipartite subgraphs
of the contact graph, and prove a cubical version of the flat plane theorem.
Similar considerations lead to a characterization of strong relative hyperbolicity of cocompactly cubulated groups relative to the hyperplane stabilizers.
Motivated by the question of which contact graphs are quasi-isometric to
bounded trees, we introduce the simplicial boundary of a locally finite cube
complex, which is a combinatorial analogue of the Tits boundary and encodes,
roughly, the cubical flat sectors in the cube complex. We establish basic properties of the simplicial boundary, and use these to characterize locally finite,
essential, one-ended cube complexes with bounded contact graph. This leads
to a reinterpretation and modified proof of the Caprace-Sageev rank-rigidity
theorem. Finally, we relate the graph metric on the 1-skeleton of the simplicial
boundary of the cube complex X to the divergence of geodesic rays in the
median graph X1 . We show that a group G acting properly, cocompactly, and
essentially on the geodesically complete cube complex X has linear divergence
function if and only if the simplicial boundary of X is connected. Otherwise,
the divergence function of G is at least quadratic; this partially generalizes a
result of Behrstock-Charney on the divergence of right-angled Artin groups.
iii
ABRÉGÉ
Nous étudions la geometrie des graphes medians et des complexes
cubiques CAT(0) en introduisant deux objets combinatoires: le graphe
de contact et la frontière simpliciale. Le premier de ces objets encode les
intersections des porteurs des hyperplans. Nous prouvons que ce graphe
est toujours quasi-isométrique à un arbre, et en déduisons que les groupes
agissant de manière propre et cocompacte sur les complexes cubiques sont
faiblement hyperboliques relativement à la famille des stabilisateurs des
hyperplans. En utilisant des techniques diagrammatiques de Casson-SageevWise, nous étudions les sous-graphes bipartis complets du graphes de contact,
et prouvons une version du théorème sur l’existence d’un plongement d’un
plan (le “flat plane theorem”). Des considérations similaires nous mènent à
une caractérisation des groupes cocompactement cubulés qui sont fortement
hyperboliques relativement aux stabilisateurs des hyperplans. Motivés par la
désir de savoir quels graphes de contact sont quasi-isométriques à des pointes,
nous introduisons la frontière simpliciale d’un complexe cubique localement
fini, qui est un analogue combinatoire de la frontière de Tits, et encode les
secteurs cubiques plats du complexe cubique. Nous établissons les propriétés
élémentaires de la frontière simpliciale, et utilisons celles-ci pour caractériser
les complexes cubiques localement finis, essentiels et à un bout, qui ont un
graphe de contact borné. Cela nous mène à une interpretation alternative du
théorème de rigidité de rang de Caprace-Sageev. Nous mettons une relation
entre la frontière simpliciale d’un complexe cubique X, et la divergence des
rayons géodésiques dans le graphe médian X1 . Nous montrons qu’un groupe
G qui agit proprement, cocompactement et essentiellement sur X, diverge de
manière linéaire si et seulement si la frontière simpliciale de X est connexe.
Cela généralise un résultat de Behrstock-Charney sur la divergence des
groupes d’Artin à angle droit.
iv
TABLE OF CONTENTS
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . .
ii
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
ABRÉGÉ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1
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6
7
9
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14
General preliminaries . . . . . . . . . . . . . . . . . . . . . .
Cubical preliminaries . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Cubes and hyperplanes . . . . . . . . . . . . . . . . .
2.2.2 Disc diagrams . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Leaves and cut-compacta . . . . . . . . . . . . . . . .
2.2.4 Sageev’s construction, cubes, and the Roller boundary
2.2.5 Restriction quotients . . . . . . . . . . . . . . . . . .
Contact and crossing graphs . . . . . . . . . . . . . . . . . .
2.3.1 Cubical products . . . . . . . . . . . . . . . . . . . .
2.3.2 Recubulation . . . . . . . . . . . . . . . . . . . . . . .
Groups acting on cube complexes . . . . . . . . . . . . . . .
2.4.1 Actions on cube complexes and contact graphs . . . .
2.4.2 Proper actions . . . . . . . . . . . . . . . . . . . . . .
2.4.3 Cocompact actions and the contact graph . . . . . . .
2.4.4 Essential actions . . . . . . . . . . . . . . . . . . . . .
2.4.5 Unambiguous actions . . . . . . . . . . . . . . . . . .
2.4.6 Examples of cubulations . . . . . . . . . . . . . . . .
14
15
15
21
23
25
30
31
35
38
40
40
41
42
45
46
48
Weak hyperbolicity and quasi-arboreal groups . . . . . . . . . . . .
52
3.1
3.2
53
57
57
60
1.2
2
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and robots
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CAT(0) cube complexes and the associated graphs
2.1
2.2
2.3
2.4
3
Background and applications . . . . . . . .
1.1.1 Cube complexes in group theory . .
1.1.2 Event structures, phylogenetic trees,
Outline and main results . . . . . . . . . .
Contact graphs are quasi-trees . . . . . .
Applications to cubulated groups . . . .
3.2.1 Weak hyperbolicity . . . . . . . .
3.2.2 Examples of quasi-arboreal groups
v
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4
3.2.3 Problems on weak hyperbolicity and quasi-arboreality
62
Bicliques in the contact graph . . . . . . . . . . . . . . . . . . . . .
64
4.1
4.2
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The simplicial boundary of a cube complex . . . . . . . . . . . . . .
89
4.3
5
5.1
5.2
5.3
5.4
6
7
Thin bicliques and the cubical flat plane theorem . . .
Hyperbolicity relative to hyperplane-stabilizers . . . . .
4.2.1 Fine contact graphs: the tiny bicliques property
4.2.2 Almost malnormality of hyperplane stabilizers .
Fractional flats and their crossing graphs . . . . . . . .
4.3.1 Fractional flats . . . . . . . . . . . . . . . . . . .
4.3.2 Nearjoins: crossing graphs of fractional flats . .
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Definition and basic properties . . . . . . . . . . . . . . . . .
5.1.1 Unidirectional boundary sets . . . . . . . . . . . . . .
5.1.2 Simplices at infinity . . . . . . . . . . . . . . . . . . .
5.1.3 Products and joins . . . . . . . . . . . . . . . . . . . .
Visible simplices and pairs of simplices . . . . . . . . . . . .
5.2.1 Maximal simplices are visible . . . . . . . . . . . . . .
5.2.2 Cube complexes are “optical spaces” . . . . . . . . . .
5.2.3 Rank one notions and fellow-traveling with hyperplanes
Comparison to other boundaries . . . . . . . . . . . . . . . .
Problems on the simplicial boundary . . . . . . . . . . . . .
89
89
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102
106
107
111
114
117
119
Bounded contact graphs and their applications . . . . . . . . . . . .
120
6.1
6.2
6.3
121
130
141
The projection trichotomy . . . . . . . . . . . . . . . . . . .
Characterizing bounded contact graphs . . . . . . . . . . . .
Rank-rigidity from the contact graph viewpoint . . . . . . .
6.3.1 Caprace-Sageev double-skewering from a contact graph
viewpoint . . . . . . . . . . . . . . . . . . . . . . .
6.3.2 One-endedness when ΓX is bounded . . . . . . . . . .
6.3.3 An alternative proof of the rank-rigidity theorem . . .
144
145
146
The combinatorial Tits boundary and divergence . . . . . . . . . .
152
7.1
7.2
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154
154
155
162
166
167
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
175
7.3
∂△ X as a combinatorial Tits boundary . . . . . . . . . . .
Divergence of pairs of combinatorial geodesic rays . . . . .
7.2.1 Bounding the divergence from above . . . . . . . . .
7.2.2 Bounding the divergence from below . . . . . . . . .
7.2.3 Divergence and the simplicial boundary . . . . . . .
Application to divergence of cocompactly cubulated groups
vi
LIST OF FIGURES
Figure
2–1 A CAT(0) cube complex and its hyperplanes. The crossing graph
is shown in the center, and the contact graph at right. These
graphs encode the contacting relations and are defined below.
17
2–2 Left to right at the top are heuristic pictures of the carriers of
a: nongon, monogon, oscugon, and bigon. Below each of these
figures is an actual disc diagram containing the corresponding
configuration. In both sets of pictures, the dual curve itself is
decorated with an arrow. . . . . . . . . . . . . . . . . . . . . .
22
2–3 The three pictures at the top show orientations of non-crossing
walls: the third is inconsistent. At the bottom: crossing walls
can be oriented independently, and the four possible orientations determine the 1-skeleton of a 2-cube. . . . . . . . . . . .
27
2–4 Turning an osculation into a crossing. . . . . . . . . . . . . . . .
40
2–5 Above is the compact quotient of X coming from the Z-action
in Example 2.4.3. Below is the part of the induced non-compact
quotient of contact graphs. . . . . . . . . . . . . . . . . . . . .
45
2–6 At left is X, with two orbits of hyperplanes. Collapsing one orbit yields a locally infinite cube complex, since there are pairs
of 0-cubes arbitrarily far apart that are separated by blue hyperplanes only. . . . . . . . . . . . . . . . . . . . . . . . . . .
47
3–1 The arrowless hyperplanes are those separating V0 from Vn . The
arrowed hyperplanes form a path in the contact graph from
V0 to Vn . Every separating hyperplane either belongs to this
path or crosses at least one of the hyperplanes on this path. .
54
3–2 At left is the graded graph Γ, partitioned into grades (indicated
by the translucent colored annuli), which are further partitioned into clusters. At right is the corresponding cluster tree.
56
3–3 The diagram D used in the cluster-tree proof of Theorem 3.1.1.
As is often convenient when considering osculations, we draw
hyperplane-carriers rather than hyperplanes. . . . . . . . . . .
57
4–1 The diagram arising from a 4-cycle in ∆X. . . . . . . . . . . . .
67
vii
4–2 The diagram Dn and a some vertical and horizontal separating
dual curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
4–3 A heuristic picture of how 4-cycles give rise to complete bipartite subgraphs of ΓX. At least one of the two colored sets of
hyperplanes is infinite, and we thus obtain K2,∞ in ΓX. . . . .
75
4–4 The three species of 3-cycle. . . . . . . . . . . . . . . . . . . . .
76
4–5 The three flavors of 4-cycle. . . . . . . . . . . . . . . . . . . . .
78
4–6 Flavor I possibilities. . . . . . . . . . . . . . . . . . . . . . . . .
78
4–7 The 4-cycle γ is neither semicondensed nor condensed at left,
semicondensed in the center, and condensed at right. The corresponding full subgraphs of Γ are shown below. There are
other possibilities, according to whether the illustrated edges
of Γ correspond to crossings or osculations. Warning: while
failure to be (semi)condensed is always reflected in some disc
diagram as shown, the converse is not true, since a separating
dual curve may map to a hyperplane that crosses one of the
labeled hyperplanes outside of the disc diagram. . . . . . . . .
80
4–8 e-reducing to a 3-cycle. . . . . . . . . . . . . . . . . . . . . . . .
81
4–9 At left is the situation arising from K2,∞ in the contact graph
of a cocompact cubulation. At right is the median argument
showing that this situation contradicts malnormality of the
hyperplanes. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
4–10 Clockwise from left: a flat, a nondiagonal half-flat, and a nondiagonal quarter-flat. . . . . . . . . . . . . . . . . . . . . . . .
85
4–11 An eighth-flat. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
4–12 A diagonal quarter-flat. . . . . . . . . . . . . . . . . . . . . . . .
86
4–13 Part of a proper weak nearjoin, arising as the crossing-graph of
an eighth-flat. . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
5–1 No facing triple implies semi-nested. . . . . . . . . . . . . . . . .
90
5–2 The map f consistently orients all hyperplanes of facing type. .
92
5–3 At left is a minimal UBS. At right is a non-minimal UBS. . . . .
93
5–4 The set of hyperplanes at left is bi-infinite. The set in the center contains a facing triple. The set of un-arrowed hyperplanes
at right is not inseparable, since the arrowed hyperplane separates two of its hyperplanes. . . . . . . . . . . . . . . . . . . .
93
viii
5–5 A typical UBS corresponding to a 1-simplex at infinity. . . . . .
99
5–6 The configuration of hyperplanes and segments arising when there
is no dual ray associated to a simplex at infinity. . . . . . . . 109
5–7 Truncation of the union of two geodesic rays. . . . . . . . . . . .
112
5–8 Folding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
113
5–9 Each of the two colored rays at left bounds an eighth-flat. At
right is part of a ray in R+ 3 bounding an eighth-flat. . . . . .
115
5–10 The top-left geodesic bounds a diagonal half-flat. The bottom
left pair of rays together bound a diagonal quarter-flat, and
each bounds an eighth-flat. The two colored rays in R+ 3 cobound a diagonal quarter-flat. . . . . . . . . . . . . . . . . . .
115
6–1 At left are the situations described in statements (2) and (3) of
Theorem 6.1.1. The assumption that γ projects to a finitediameter path in Γ leads to one of the situations (4) or (5)
shown at right. . . . . . . . . . . . . . . . . . . . . . . . . . .
122
6–2 The diagram Di in the proof of the first assertion of the projection trichotomy. . . . . . . . . . . . . . . . . . . . . . . . . . .
123
6–3 Each dual curve, or configuration of dual curves, is impossible.
The numbered configurations are illustrated in the next two
figures; the unnumbered ones contradict minimality of Di or
ψi for straightforward reasons. . . . . . . . . . . . . . . . . . .
124
6–4 Di does not contain “triangles along the boundary”. The singlearrowed path precludes crossing dual curves emanating from
the same “syllable” of Pi , while the double-arrowed path precludes dual curves starting and ending on consecutive syllables. 124
6–5 The dual curves in Di . . . . . . . . . . . . . . . . . . . . . . . .
125
6–6 The diagram Ei is shown at left inside Di , and at right with the
dual curves named in the proof of Theorem 6.1.1. . . . . . . .
127
6–7 If ∆X is disconnected, then X has a cut-0-cube. . . . . . . . . .
132
6–8 The red and green hyperplane osculate but the shortest path
joining them in ∆X has length 7. A disc diagram argument
shows that the intersection of their carriers is a vertex of degree at least 8. . . . . . . . . . . . . . . . . . . . . . . . . . .
133
ix
6–9 An infinite spiral of quarter-flats. The red hyperplane is at distance 5 from the blue one in ∆X, and the simplicial boundary
of the part of X pictured is a subdivides interval of length 5.
The arrowed geodesic spends finite time in each quarter-flat,
and represents a pair of isolated simplices in ∂△ X. . . . . . . .
134
7–1 The sets Ar , Br , Cr are shown at left. At right is the path Qr . . .
157
7–2 A heuristic picture of the fan decomposition of a shortest r-avoiding
path from γ(r) to γ ′ (r). . . . . . . . . . . . . . . . . . . . . . 162
7–3 The diagram Dr showing that either η(v, v ′ ) ≤ 1 or ur → ∞. . .
x
164
CHAPTER 1
Introduction
1.1
Background and applications
Nonpositively-curved cube complexes are remarkable in that their
definition seems somewhat artificial – at first sight, they are just a specific
type of CAT(0) piecewise-Euclidean polyhedral complex – but they have a
surprisingly rich geometric and combinatorial nature, and arise naturally in
geometric group theory, low-dimensional topology, and discrete mathematics.
The goal of this thesis is to describe the manner in which many geometric
properties of cube complexes are controlled by purely combinatorial objects.
We shall introduce two natural structural invariants of a CAT(0) cube
complex, the contact graph and the simplicial boundary, which encode
properties of the cubical structure but ignore the CAT(0) geometry, and
exhibit several large-scale properties of the cube complex which can be
inferred from simple properties of the contact graph and the simplicial
boundary. We shall also explore the relationship between these two objects,
and apply our findings to groups acting on cube complexes.
From the point of view of CAT(0) geometry, cube complexes are metric
polyhedral complexes of nonpositive curvature whose cells are Euclidean
cubes of various dimensions. It is natural to construct complexes whose cells
are Euclidean (hyperbolic, spherical) polyhedra, glued together along their
faces by isometries, and expect that the Euclidean (hyperbolic, spherical)
metrics on the cells fit together to yield a geodesic metric on the whole
space. Indeed, Bridson showed that a simply-connected, piecewise-Euclidean
polyhedral complex with finitely many isometry types of cells supports a
piecewise-Euclidean geodesic metric [Bri91]. Gromov had already given a
purely combinatorial criterion, in terms of the links of vertices, guaranteeing
1
that such a polyhedral complex is nonpositively curved (more precisely,
CAT(0)). When the complex in question is assembled from Euclidean unit
cubes of bounded dimension, it is somewhat simpler to verify the existence
of a piecewise-Euclidean metric, and this, together with the link criterion,
led to the introduction of CAT(0) cube complexes, as a source of examples,
in [Gro87]. Leary has since shown that any simply connected cube complex
satisfying the link condition is CAT(0) (regardless of whether the dimension is
finite) [Lea10].
However, cube complexes have a large amount of additional structure,
to the extent that restricting oneself to the CAT(0)-geometric viewpoint
when studying cube complexes is something of a handicap. This additional
geometry comes from the hyperplanes in a CAT(0) cube complex, and yields
so much information that we shall adopt the hyperplane-centric viewpoint
in which a CAT(0) cube complex is by definition a simply connected cube
complex satisfying Gromov’s link criterion; we shall largely ignore the CAT(0)
metric.
The hyperplane-dictated geometry of cube complexes became apparent
in the work of Sageev on semi-splittings of groups. The story begins with the
Bass-Serre theorem, which says that the group G splits nontrivially as a graph
of groups if and only if G admits a nontrivial isometric action on a tree with
no edge-inversions. To generalize this result requires a generalization of the
notion of a tree, and a generalization of the notion of a splitting.
The latter came first, in work of Houghton and Scott, departing from
a famous result of Stallings. The finitely generated group G, with some
word-metric, has a space of ends, whose cardinality e(G) ∈ {0, 1, 2, ∞} is
independent of the choice of (finite) generating set. Stallings showed that, if
e(G) > 1, then G admits a splitting over a finite subgroup, i.e. G acts on a
tree with finite edge stabilizers. If H ≤ G is a subgroup, one can define the
number e(G, H) of relative ends to be the number of ends of the quotient of a
2
Cayley graph of G by the action of H (see [Hou74, Sco]). If e(G, H) > 1, then
G admits a semi-splitting over H.
Sageev showed that the existence of a semi-splitting of G is equivalent to
the existence of an action of G on a CAT(0) cube complex, generalizing the
Bass-Serre theorem. The CAT(0) cube complex given by Sageev’s theorem
plays the role of the Bass-Serre tree, and this is a natural generalization
since the class of 1-dimensional CAT(0) cube complexes is exactly the class
of simplicial trees. The relevant property of a tree T is the fact that, for
each edge e, with midpoint m(e), the space T − {m(e)} has exactly two
components. Similarly, the CAT(0) cube complex X contains hyperplanes:
a hyperplane H is a convex subspace, intersecting each 1-cube of X in its
midpoint, with the property that X − H has exactly two components, called
halfspaces [Sag95]. The stabilizers of the hyperplanes are codimension-1
subgroups of G and function as “generalized edge groups”.
Hyperplanes govern the geometry of a CAT(0) cube complex via a
combination of their convexity and their separating properties. In discrete
mathematics, another class of spaces, generalizing trees and containing many
convex, separating subspaces, was already the subject of a great deal of
study: median spaces. In addition to the fact that midpoints of edges are
separating, trees have the median property: given three distinct vertices x, y, z,
there exists a unique vertex m = m(x, y, z), such that each of the pairs
{x, y}, {x, z}, {y, z} is joined by a geodesic segment containing m, so that
d(x, y) = d(x, m) + d(m, y)
d(y, z) = d(y, m) + d(m, z)
d(z, y) = d(z, m) + d(m, x).
A metric space with this property is a median space, and a graph which, when
equipped with the standard path metric, has this property is a median graph.
Median graphs have been studied extensively (see, e.g. the surveys [BC08,
BH83, Isb80]) and are intimately related to cube complexes. The relationship
3
arises from a consequence of the median property: a convex subgraph S of a
median graph M is gated, in the sense that, for any vertex v of M , there is a
unique closest vertex of S to v. One defines a convex split in M to be a pair
(A, M 0 − A), where A is a convex vertex-set for which M − A is also convex.
The halfspaces A and M 0 − A are both gated, and one can use this to define
the Djoković-Winkler relation on the set of edges: two edges {x, y}, {u, v}
are equivalent if there is a convex split separating x from y and u from v.
On the other hand, if X is a CAT(0) cube complex, the hyperplanes yield a
very similar relation on the 1-cubes: two 1-cubes are equivalent if both are
dual to the same hyperplane H, where the 1-cube c is dual to H if H contains
the midpoint of c [Sag95]. This is no accident, because, in [Che00], Chepoi
showed that the class of median graphs is precisely the class of 1-skeleta
of CAT(0) cube complexes: the 1-skeleton of a CAT(0) cube complex is a
median graph, and, given a median graph, one may add higher-dimensional
cubes “wherever their 1-skeleta appear” to obtain a uniquely-determined
CAT(0) cube complex. In addition to median spaces, there are also median
algebras, and the relationship between median algebras, median graphs, cube
complexes, poc-sets and Sageev’s construction of a cube complex from a group
semi-splitting was explored by Roller in [Rol98].
To place Sageev’s construction in a more general context, one must
abstract the salient property of the codimension-1 subgroup H ≤ G – the
subgroup with respect to which G has more than one end – namely that
H stabilizes a certain bipartition of G. The correct setting is a wallspace,
introduced by Haglund and Paulin [HP98]. A wallspace is a set together with
a collection of designated bipartitions of the underlying set into halfspaces –
these bipartitions are the walls – subject to a “finite interval condition” saying
that any two elements of the underlying set are separated by finitely many
walls. There are then several essentially similar ways to construct the cube
complex dual to the wallspace. One approach, due to Nica, holds that 0-cubes
are to be viewed as principal ultrafilters on the collection of halfspaces [Nic04].
4
Nica shows that the 0-skeleton, thus constructed, is a discrete median algebra
and therefore the 0-skeleton of a CAT(0) cube complex. The approach taken
by Chatterji-Niblo in [CN05], and adopted in the present document, defines
each 0-cube to be a way of simultaneously orienting all of the walls, subject to
the conditions that (1) no two walls may be “pointed away from one another”
and (2) any 0-cube differs on finitely many walls from an orientation of all
walls “toward” an element of the underlying set. Each of these constructions
yields a median graph; one then either fills in the higher-dimensional cubes, or
appeals to Chepoi’s result.
In view of this construction, a CAT(0) cube complex canonically determines a wallspace: the walls are the partitions of the 0-skeleton induced by
the hyperplanes, and performing the above construction on this wallspace
returns the original cube complex. In this way, one can think of the 0-cubes
of a CAT(0) cube complex as simultaneous orientations of all hyperplanes,
subject to conditions (1) and (2). This yields a nice interpretation of the standard path-metric on the 1-skeleton: the geodesic distance between two 0-cells
is equal to the number of hyperplanes that separate them. A median graph
thus becomes a geodesic space in which geodesic segments are completely
characterized by the property that they never cross the same hyperplane
twice.
In this thesis, the geometry we shall study is really that of median
graphs. Our object of investigation is a CAT(0) cube complex X, but metrically, we are interested in the median graph X1 with its standard path-metric.
This metric actually extends to a metric on X, whose restriction to each cube
is the ℓ1 metric, and many metric properties of subcomplexes (notably, convexity) hold regardless of whether one considers this “hyperplane ℓ1 metric”
or the piecewise-Euclidean CAT(0) metric [Hag07]. However, we shall take an
entirely combinatorial viewpoint, focusing on the hyperplanes, retaining the
higher-dimensional cubes largely because they permit the use of disc diagrams.
5
Casson observed that certain combinatorial homotopies, called hexagon
moves, are available in disc diagrams whose 2-cells are 2-cubes and which
map to spaces satisfying the link condition. Disc diagram techniques for cube
complexes were subsequently developed by Sageev in his thesis, and used
in [Sag95, Sag97]. In [Che00], Chepoi proved several results about, and made
extensive use of, disc diagrams. The theory was greatly extended by Wise
in [Wisa], in the context of small-cancellation theory over cube complexes.
We shall make extensive use of disc diagrams, following procedures in [Wisa]
and [Hag11].
1.1.1
Cube complexes in group theory
Finding codimension-1 subgroups in groups of a given class, and thereby
constructing actions on cube complexes, has become a successful industry,
and in Chapter 2, we mention many classes of groups, each of which occurs
naturally in geometric group theory for independent reasons, that act nontrivially on CAT(0) cube complexes. By virtue of the existence of such an
action, these groups enjoy numerous properties: for example, they do not
have Kazhdan’s property (T) [NR98], they can often be shown to have finite
asymptotic dimension [Wri10, Hag11] and the rapid decay property [CR05],
they are often satisfy rank-rigidity (see [CS11] and Chapter 6 of the present
document), many satisfy the Tits alternative [SW05, CS11], they generally
admit many pseudocharacters [CS11, BC11], and are weakly amenable [GH10].
In Chapter 3, and in [Hag11], it is shown that a group acting on a CAT(0)
cube complex acts on a quasi-tree, and we use this to discuss weak and strong
relative hyperbolicity of such groups. This quasi-tree is the contact graph of
the cube complex, which is one of the central objects of study in this thesis.
Members of the large class of groups with quasiconvex hierarchies, which
includes hyperbolic limit groups and fundamental groups of Haken hyperbolic
3-manifolds, have been shown in recent work of Wise to act on CAT(0) cube
complexes in such a way that the quotient is (virtually) special, in the sense
defined by Haglund and Wise [HW08, Wisa]. Such groups are characterized by
6
their status as subgroups of right-angled Artin groups [HW08], and thus have
additional useful properties. Notably, they are residually finite Q-solvable, in
the sense of [Ago08], and this fact plays a crucial role in Wise’s proof that,
for closed hyperbolic 3-manifolds (and more), the virtual fibering and virtual
Haken conjectures are equivalent.
1.1.2
Event structures, phylogenetic trees, and robots
Cube complexes and median graphs also arise outside of group theory;
the author suspects that their seeming ubiquity is due to the extremely
natural notion of a wallspace. One such application of cube complexes is
to event structures with binary conflict. These objects arise in the study of
concurrency in computer science (see e.g. [Win82]). Event structures are
discrete objects modeling concurrent computation, and were first studied by
Winskel in his PhD thesis and by Nielsen-Plotkin-Winskel in [NPW85]. An
event structure is a set E with a partial ordering called causal dependence and
a binary, irreflexive, symmetric relation called conflict, with the property that
causal dependence preserves conflict. We also require that causal dependency
satisfies the finite interval condition. Two non-conflicting events that are
incomparable in the partial ordering are concurrent. Concurrency is the
simpler of the two ways in which events can be independent; the other is
minimal conflict. In [RT91], Rozoy and Thiagarajan asked for conditions
under which an event structure admits a nice labeling, i.e. a map from E to
some finite set such that any two independent events have distinct images.
From an event structure, one can construct a domain, which is a kind
of “configuration space”: the points of the domain are subsets C of E, called
configurations, such that the elements of C are pairwise non-conflicting, and
C is closed under non-conflicting causal dependence. A configuration should
be thought of as a “potential history” of a system, so that the occurrence of a
single possible event – i.e. the addition of e ∈ E to C – moves the system to a
new configuration; see [WN95]. Beautifully, the graph whose vertex set is the
set of configurations, with two configurations joined by an edge exactly when
7
they differ by a single event, is a median graph [BC93], and hence the domain
of an event structure is a based CAT(0) cube complex whose hyperplanes
correspond to events. Independence, causal dependence, and conflict have
simple interpretations in terms of intersections of hyperplanes and their
cubical neighborhoods (explicit descriptions appear in [Che11, CH11]).
In [Che11], Chepoi constructed a counterexample to the nice labeling
conjecture, by exhibiting an event structure whose domain, a 4-dimensional,
finite-degree CAT(0) cube complex, has a contact graph with infinite chromatic number. Chepoi’s example is based on a construction of Burling [Bur65]
which, given n ≥ 0, more or less produces a wallspace whose contact graph
does not admit a proper coloring with n colors. Chepoi shows that to construct a nice labeling of an event structure, one must properly color a certain
subgraph – the pointed contact graph – of the contact graph of its domain,
and it follows that the event structure of which Chepoi’s cube complex is the
domain does not admit a nice labeling. In [CH11], Chepoi and the author
employed the recubulation construction described in Chapter 2 to produce a
5-dimensional cube complex, based on Chepoi’s example, that does not embed
in the product of finitely many trees. On the other hand, we also showed that
the contact graph of a finite-degree, 2-dimensional CAT(0) cube complex has
chromatic number bounded by a polynomial function of the degree, and thus
such a complex embeds in a finite product of trees. This addresses a question
of Sageev, which he answered positively in the hyperbolic case and, with
Druţu, in the relatively hyperbolic case.
Cube complexes have also arisen in evolutionary biology, in the study
of phylogenetic trees. Phylogenetic trees model the evolutionary history
of a given collection of organisms; they are rooted directed trees whose
vertices correspond to “operational taxonomic units”, for example species,
populations, or genes. Adjacency corresponds to ancestry in the appropriate
sense, and closeness of two vertices corresponds to relatively recent common
ancestry of the corresponding taxonomic units. Having fixed the leaves, the
8
idea is to infer the structure of the corresponding phylogenetic tree from
the morphological similarities of the corresponding taxonomic units. In
other words, a fixed collection of organisms has many potential evolutionary
histories, and each candidate history is represented by a tree. In [BHV01],
Billera, Holmes, and Vogtmann construct a “space of trees” with a fixed set
of leaves, and study the geometry of this space. This allows one to compare
candidate trees. The tree space is a CAT(0) space built from Euclidean
orthants, and as such enjoys a CAT(0) cubical structure.
Cube complexes arise in robotics. There are various formalizations
of what it means to be a robot (see e.g. [GP05, AG02]), but a common
feature, and one possessed by the metamorphic robots discussed by AbramsGhrist-Peterson, is that a robot may be, at any time, in one of a given set
of configurations, and may change configuration by performing one of a
fixed set of moves. Appropriately formalized, this configuration space is a
nonpositively-curved cube complex [GP05, AG02]. A natural question, for
example, about a robot is: from a given state, what sequence of moves allows
the robot to pass most efficiently to another given state? Recently, ArdilaOwen-Sullivant have produced an algorithm that finds geodesics in a CAT(0)
cube complex, given the endpoints [AOS11], and applied this result to this
question about metamorphic robots.
1.2
Outline and main results
In Chapter 2, we give the necessary background. Basic properties of
cube complexes are discussed in Section 2.2. In Section 2.3, we describe the
crossing graph and the contact graph of a cube complex. This section is
largely devoted to interpreting cubical subcomplexes, quotients, and products
in terms of the contact graph. The remainder of the chapter is devoted to
group actions on cube complexes and their induced actions on the contact
graph.
Chapter 3 contains a simple proof of the main result of [Hag11], which
says that the contact graph of any CAT(0) cube complex is quasi-isometric
9
to a tree, as well as a sketch of the original proof, using disc diagrams1 .
We recall Bowditch’s and Farb’s definitions of a weakly hyperbolic group,
focusing on the case of a group acting properly and cocompactly on a cube
complex in which the peripheral subgroups are the hyperplane-stabilizers. We
verify that the induced action on the contact graph satisfies the criteria of
Bowditch’s definition, and thus that the group is weakly hyperbolic relative
to the collection of hyperplane stabilizers. We conclude with some examples
of groups that are weakly hyperbolic by virtue of acting on quasi-trees, but
which are not cubulated.
Bowditch’s fine graph condition for (strong) relative hyperbolicity is
taken up in Chapter 4. First, we establish a cubical version of the flat plane
theorem using disc diagram techniques, following [Hag11]: a group acting
properly and cocompactly on a CAT(0) cube complex is word-hyperbolic if
and only if the crossing graph does not contain K∞,∞ . By related methods,
we show that a group acting properly, cocompactly, and unambiguously on a
CAT(0) cube complex is hyperbolic relative to the collection of hyperplanestabilizers if and only if that collection is almost malnormal. This is proved
by showing that the contact graph fails to be fine if and only if it contains the
complete bipartite graph K2,∞ . It follows from the cubical flat plane theorem
that, if the contact graph is fine, then the group in question is already wordhyperbolic.
The conditions on bicliques in Chapter 4 motivate the definition of the
simplicial boundary of a cube complex given in Chapter 5, which is devoted
to defining this boundary and elucidating some of its properties. Roughly,
the simplicial boundary is a “simplicial complex at infinity” whose positivedimensional simplices keep track of unbounded cubical flat regions in the cube
complex; in particular, the simplicial boundary is totally disconnected if the
1
The latter proof, given in [Hag11], yields much better quasi-isometry constants, at the expense of being considerably more complicated.
10
cube complex is hyperbolic. The initial definition of the simplicial boundary
is given in terms of sets of hyperplanes that are closely related to the 0-cubes
at infinity discarded when performing Sageev’s construction. Section 5.2
relates the simplicial boundary to the space of hyperplane-equivalence classes
of combinatorial rays. The chapter concludes with a series of examples that
distinguish the simplicial boundary from other extant boundaries: the visual
boundary, the Tits boundary, and the Roller boundary.
The simplest tree is a point, and the simplest quasi-tree, from that point
of view, is a quasi-point. The fact that contact graphs are quasi-isometric to
trees motivates the following question, which in turn provided the impetus for
the development of the simplicial boundary: under what geometric conditions
does the contact graph have finite diameter? We answer this in Chapter 6.
The main tool is Theorem 6.1.1, which describes the only two ways in which
a combinatorial geodesic ray can fail to project to an unbounded ray in the
contact graph: it can lie in an infinite combinatorial flat “sector”, or it can
fellow-travel with a hyperplane. Using this trichotomy, we deduce that the
contact graph of an essential, one-ended, locally finite CAT(0) cube complex
is bounded if and only if the 1-skeleton of the simplicial boundary is bounded;
we actually give bounds on the ratio of the diameters of these graphs. Each of
these conditions is in turn equivalent to the statement that the cube complex
has a pseudo-product structure, which somewhat artificially generalizes the
state of being a cubical product.
Using similar ideas, we give a modified statement and proof of the rankrigidity theorem recently proved by Caprace-Sageev [CS11]. Specifically,
we show that if G acts properly, essentially, and cocompactly on the cube
complex X, then either there exists g ∈ G that has a quasi-geodesic axis
in the contact graph, in which case g is a rank-one element, or the contact
graph is bounded, in which case X splits as a nontrivial product. The
trichotomy theorem described above is also used to give a kind of converse
to the rank-rigidity theorem: if G acts properly and cocompactly on X, then
11
each rank-one element either has a quasi-geodesic axis in the contact graph, or
has a power that stabilizes a hyperplane.
In Chapter 7, we re-interpret the simplicial boundary as a “combinatorial
Tits boundary” of a cube complex. Some CAT(0) notions don’t have natural
counterparts in the world of median graphs, largely because metric balls
in a median graph need not be convex. Nonetheless, a “combinatorial Tits
(pseudo)metric” can be defined, and we conclude the chapter by relating this
metric to the divergence of geodesics in a CAT(0) cube complex, discussed
and defined in [Ger94b, Ger94a, BC11, DMS10], among other places. This
yields a generalization of a recent result of Behrstock-Charney: they proved
that a one-ended right-angled Artin group has linear or quadratic divergence,
according to whether or not the generating graph is a nontrivial join. We
show that a group G acting properly, essentially, and cocompactly on a
CAT(0) cube complex that is combinatorially geodesically complete has linear
divergence if and only if the simplicial boundary of the complex is a nontrivial
join (i.e. if and only if the contact graph is bounded). Otherwise, G has at
least quadratic divergence.
Mandatory originality statement: This thesis is an account of the
author’s work of the past two years, some of which has already been submitted for publication. However, things have been rewritten and reorganized
considerably. Some parts of this manuscript are reworked or taken verbatim
from the author’s paper [Hag11]; this has been indicated where it occurs. In
particular, the material in Chapter 3 and the first section of Chapter 4 is
contained in [Hag11], although it has been rewritten in a more streamlined
way here. The results in the second half of Chapter 4 have not been submitted
for publication at the time of writing. The results in Chapter 2 that are not
accompanied by citations are proved in [Hag11], although we have reproduced
some of the proofs here. These tend to be fairly routine applications of the
definitions. All of the results in this thesis are the work of the author, unless
noted otherwise. Chapters 5, 6, and 7 were written concurrently with the
12
author’s paper [Hag12], which contains the results of these chapters and which
was submitted for publication while the initial version of this thesis was under
examination. Where we have re-proved pre-existing results, there is discussion
of the differences and similarities between the proofs.
13
CHAPTER 2
CAT(0) cube complexes and the associated graphs
This chapter’s description of CAT(0) cube complexes and their basic
features partially follows the synopses in [CH11, Hag11, Wisa] and defines
most of the objects and many of the techniques used throughout.
2.1
General preliminaries
The definitions and terminology related to metric spaces, isometries,
geodesics, geometric group actions, quasi-isometries, hyperbolicity, and
CAT(0) spaces employed in this document are consistent with that in the
standard reference, [BH99].
The basic material on CW-complexes and simplicial complexes used
here can be found in textbooks like [Hat02]; we use the terms “combinatorial
map” and “cubical map” as they are used in [Wisa]. Unless stated otherwise,
graphs are simplicial – each edge has two distinct vertices, and no two vertices
determine more that one edge – and are equipped with the standard path
metric in which each edge has unit length.
We often appeal to the following graph-theoretic version of the axiom of
choice:
Proposition 2.1.1 (König’s lemma [Kön36]). Let Λ be a locally finite graph.
Then every infinite component of Λ contains an infinite ray.
If Γ1 , Γ2 are graphs, the join Γ1 ⋆ Γ2 is the graph obtained from Γ1 ⊔ Γ2
by adding all edges of the form (v1 , v2 ), where v1 ∈ Γ1 and v2 ∈ Γ2 . Likewise,
if K1 , K2 are simplicial complexes, then the simplicial join K1 ⋆ K2 of K1
and K2 is the simplicial complex obtained from K1 ⊔ K2 by adding a simplex
[k11 , . . . , k1n , k21 , . . . , k2m ] for each pair of simplices [k11 , . . . , k1n ] ⊆ K10 and
[k21 , . . . , k2m ] ⊆ K20 . Note that if K1 , K2 are flag complexes, then K1 ⋆ K2 is the
unique flag complex whose 1-skeleton is the graph K11 ⋆ K21 .
14
If V1 , V2 are totally disconnected graphs (i.e. vertex-sets), then the
biclique K(V1 , V2 ) = V1 ⋆ V2 . The sets V1 , V2 are the sides of the bipartition
of K(V1 , V2 ). If we only care about the cardinalities n = |V1 |, m = |V2 |, then
we use the notation Km,n = K(V1 , V2 ). We use the notation KZ,Z , KN,Z , etc.
when speaking of a countably infinite biclique with a fixed total order on the
vertices in each side of the biclique.
2.2
Cubical preliminaries
2.2.1
Cubes and hyperplanes
Definition 2.2.1 (Cube complex). For 0 ≤ d < ∞, a d-cube (or cube of
dimension d) is a Euclidean cube [− 12 , 12 ]d . For 0 ≤ k ≤ d − 1, a k-face c′ of
the d-cube c is a subspace obtained by restricting exactly d − k coordinates
( d )
to ± 12 . Each k-face is a k-cube, and c has 2d−k d−k
distinct k-faces. A cube
complex X is a CW-complex whose cells are cubes of various dimensions, with
the property that for all cubes c, c′ of X, c ∩ c′ is a common face of c and
c′ , i.e. cubes are glued to X via combinatorial isometries of their faces. The
dimension dim X of X is the supremum of the set of dimensions of cubes of
X. The degree of X is the supremum over all 0-cubes x ∈ X of the degree of x
as a vertex of the graph X1 .
For each 0-cube x ∈ X, the link lk(x) is a simplicial complex with a
(d − 1)-simplex s(c) for each d-cube c of X that properly contains x. The
simplices s(c), s(c′ ) are attached along the face s(c′′ ) corresponding to the
common face c′′ of c and c′ . If lk(x) is a flag complex for each x ∈ X0 ,
then X is nonpositively curved. X is a CAT(0) cube complex if it is both
simply-connected and nonpositively-curved.
Note that this terminology is somewhat misleading, since it makes no
reference to a CAT(0) metric. The alternative term cubing is sometimes used,
and perhaps preferable, but we use the term “CAT(0) cube complex”, defined
in exactly this way. CAT(0) cube complexes were introduced by Gromov as
a source of examples; he showed that the finite-dimensional CAT(0) cube
15
complex X admits a CAT(0) metric [Gro87], and this was extended to infinitedimensional CAT(0) cube complexes by Leary [Lea10].
The fundamental feature of the CAT(0) cube complex X is its collection
of hyperplanes, which were identified by Sageev [Sag95].
Definition 2.2.2 (Hyperplane, carrier, separation, dual 1-cube). A midcube
c′ in the (d + 1)-cube c is a codimension-1 subspace obtained by restricting
exactly one coordinate to 0; c′ is isometric to each codimension-1 face of
c. A hyperplane H in the CAT(0) cube complex X is a connected union of
midcubes of cubes of X such that for each closed cube c, either H ∩ c = ∅ or
H ∩ c is a single midcube of c.
The carrier N (H) of the hyperplane H is the union of all closed cubes
c such that H intersects c in a midcube. In [Sag95], it is shown that H is 2sided in the sense that N (H) ∼
= [− 12 , 12 ] × H. It is also shown that H separates
X: the complement X − H has exactly two components H + and H − , called
halfspaces.
If H is a hyperplane in X and A, B are subspaces of X, then H separates
A from B if A and B lie in distinct halfspaces of H. In particular, if c is a
1-cube, then there is a unique hyperplane H that separates the endpoints
of c; this is described by saying that H is dual to c and c is dual to H. The
relation “dual to the same hyperplane” on the set of 1-cubes is an equivalence
relation upon which we shall elaborate below.
Sageev also showed that each hyperplane H of X is itself a CAT(0)
cube complex, and that H and N (H) are convex in X with respect to the
piecewise-Euclidean metric.
We shall always assume X has countably many 0-cubes. Hence X has
countably many cubes of each dimension. Since there are countably many
1-cubes, there are countably many hyperplanes.
Definition 2.2.3 (Contacting). Let H, H ′ be distinct hyperplanes of the
CAT(0) cube complex X. Then H and H ′ contact, denoted H ⌣
⊥ H ′ , if no
16
hyperplane H ′′ separates H from H ′ . Equivalently, H ⌣
⊥ H ′ if and only if
N (H) ∩ N (H ′ ) ̸= ∅.
H and H ′ can contact in one of two ways. If H ∩ H ′ ̸= ∅, then H and
H ′ cross, denoted H⊥H ′ . In this case, N (H) ∩ N (H ′ ) contains 1-cubes h
and h′ , dual to H and H ′ respectively, such that h and h′ have a common
0-cube and the concatenation hh′ forms part of the attaching map of a 2-cube.
Equivalently, H⊥H ′ if and only if each of the quarter-spaces H ± ∩ (H ′ )± ̸= ∅.
If H ⌣
⊥ H ′ and H does not cross H ′ , then H and H ′ osculate: no
hyperplane separates H from H ′ , and each of H and H ′ lies in a single
halfspace of the other. In this case, we again have a concatenation hh′ ⊆
N (H) ∩ N (H ′ ) with h dual to H and h′ dual to H ′ , but hh′ does not lie on the
boundary path of a 2-cube of X.
Figure 2–1: A CAT(0) cube complex and its hyperplanes. The crossing graph
is shown in the center, and the contact graph at right. These graphs encode
the contacting relations and are defined below.
Definition 2.2.4 (Median graph). The graph Γ is a median graph if for
each triple x, y, z of distinct vertices in Γ, there exists a unique vertex
m = m(x, y, z), called the median of x, y, z such that
d(x, y) = d(x, m) + d(y, m),
where d is the standard path-metric on Γ, and the analogous equality holds for
the triples x, z, m and y, z, m.
17
Chepoi showed in [Che00] that the class of median graphs is precisely
the class of 1-skeleta of CAT(0) cube complexes; essentially the same fact
was proved independently by Roller. Roller actually showed that the class
of 0-skeleta of CAT(0) cube complexes coincides with the class of discrete
median algebras (Theorem 10.3 of [Rol98]), and the latter class corresponds
bijectively to the class of median graphs, by a theorem of Avann [Ava61].
The relation on the set of 1-cubes defined by: c and c′ are equivalent if
they are dual to the same hyperplane is an equivalence relation and coincides
with the Djoković-Winkler relation on the edges of the median graph X1 ,
which is the transitive closure of the relation: c and c′ are related if they are
opposite sides of the same 4-cycle [EFO07, IK00]. In this way, one can verify
that the hyperplanes of X correspond to the convex splits of the median graph
X1 [Mul80, vdV93]. Another characterization of CAT(0) cube complexes
follows from this fact. Indeed, Proposition 2.3.5 below shows that, given
the CAT(0) cube complex X, there is a CAT(0) cube complex Y, whose
hyperplanes correspond bijectively to those of X, such that the hyperplanes
of Y pairwise cross and X embeds isometrically in Y. By Proposition 2.3.8,
Y is a (possibly infinite) cube. Using Sageev’s construction, one constructs a
cubical retraction map Y → X. This is a rough explanation of the fact, due
to Bandelt, that median graphs are exactly the graphs arising as retracts of
1-skeleta of (possibly infinite-dimensional) cubes [Ban84].
Almost all of our geometric considerations really concern median graphs,
though cubical techniques are so useful that we will adopt a viewpoint in
which we work inside of a CAT(0) cube complex, restricting ourselves to
combinatorial maps, paths, and so forth. This is also the approach taken
in [Wisa].
Remark 2.2.5 (Metrics on cube complexes and median graphs). Let X be
a CAT(0) cube complex. Given x, y ∈ X0 , let dX (x, y) be the length of a
minimal length path in X1 joining x to y. Equivalently, dX (x, y) counts the
hyperplanes separating x from y. There is a natural extension of this metric to
18
the whole cube complex X, which restricts to the ℓ1 metric on each cube, but
this is unnecessary for our purposes.
As mentioned before, X is endowed with a piecewise-Euclidean CAT(0)
metric dX with which each d-cube is a Euclidean unit cube. We shall almost
never use this metric, and when we do, it will be in a context in which we can
apply the following fact, which seems to be well known by those working in
the field and which is proved in [CS11]: if X is a finite-dimensional CAT(0)
cube complex, then the inclusion
(X1 , dX ) ,→ (X, dX )
is a quasi-isometry, i.e. a finite-dimensional CAT(0) cube complex is quasiisometric to its 1-skeleton.
The CAT(0) cube complex X is locally finite if each 0-cube has finite
degree. Note that a locally finite cube complex is locally compact, and X is
a proper metric space. If the degree of X is finite, then X is uniformly locally
finite, and is therefore finite-dimensional. If Y ⊆ X is a subcomplex, then
Y is isometrically embedded (respectively convex, bounded ) if Y1 ⊆ X1 is
isometrically embedded (respectively convex, bounded). For each r ≥ 0, we
denote by Nr (Y) the smallest subcomplex of X containing all 0-cubes x ∈ X
such that dX (x, Y0 ) ≤ r. This is the cubical r-neighborhood of Y. Similarly,
the convex hull of Y is the smallest convex subcomplex of X containing Y.
If Y ⊂ X is isometrically embedded, then for each hyperplane H of X,
the subcomplex N (H) ∩ Y is connected, as can be shown by a simple disc
diagram argument using the characterization of geodesic segments below. If Y
has nonempty intersection with both of the halfspaces associated to H, then
H crosses Y. Convexity can be understood in terms of hyperplanes, as well:
the following two facts are well known.
Proposition 2.2.6 (Convexity and the Helly property). Let Y be a subcomplex of X. Then Y is convex if and only if, for all hyperplanes H, H ′ that
19
cross Y, either N (H) ∩ N (H ′ ) = ∅ or N (H) ∩ N (H ′ ) ∩ Y ̸= ∅, and, if H⊥H ′ ,
then Y contains some 2-cube whose 1-cubes are dual to H, H ′ .
Hence X enjoys the Helly property: if Y1 , Y2 , . . . , Yn are convex subcom∩
plexes of X such that Yi ∩ Yj ̸= ∅ for i ̸= j, then ni=1 Yi ̸= ∅.
In particular, hyperplane carriers are convex, and each family of d
pairwise-crossing hyperplanes is dual to at least one d-cube. Proposition 2.2.6
says that Y ⊆ X is convex if it is locally-convex, in the sense that the
inclusion Y ,→ X is a local isometry. A local isometry of nonpositively-curved
(not necessarily CAT(0)) cube complexes is a locally injective cubical map
ϕ : Y → X such that, if c, c′ are 1-cubes of Y with a common 0-cube, and
ϕ(c) ∪ ϕ(c′ ) forms the corner of a 2-cube in X, then c ∪ c′ forms the corner
of a 2-cube in Y. Equivalently, ϕ is a local isometry if the induced map
lk(y) → lk(ϕ(y)) embeds lk(y) as a full subcomplex for each y ∈ Y0 .
From now on, X denotes a CAT(0) cube complex, W denotes the set of
hyperplanes in X, and for each isometrically embedded subcomplex Y, the set
of hyperplanes that cross Y is denoted W(Y ).
Definition 2.2.7 (Combinatorial paths). An interval I is a CAT(0) cube
complex obtained from [− 12 , n − 12 ], for some n ∈ N ∪ {0}, or from R≥− 1 ,
2
or from R, by subdividing so that 0-cubes are points of the form m −
1
2
for
m ∈ Z. Denote by R+ the CAT(0) cube complex obtained from R≥− 1 and by
2
R the CAT(0) cube complex obtained from R.
A combinatorial path P → X is a combinatorial map P : I → X1 , where
I is an interval. When the context is clear, the term “path” always means
“combinatorial path”, and if P is an embedding, we will denote by P both the
map and its image in X.
If I is a finite interval, then |P | = n is the length of P. If P → X is an
isometric embedding, then P is a geodesic segment in X, and |P | is equal to
the number of hyperplanes separating the endpoints of P, i.e. |P | counts the
hyperplanes in W(P ). If I = R+ , then P is a (combinatorial) geodesic ray,
and P is a (bi-infinite, combinatorial) geodesic if I = R . The path P is a
20
geodesic (geodesic ray, geodesic segment) if and only if P contains exactly one
1-cube dual to each hyperplane.
2.2.2
Disc diagrams
In a few places, we will use the technique of minimal-area disc diagrams
in non-positively curved cube complexes. This subsection summarizes parts of
the discussion of disc diagrams in CAT(0) cube complexes appearing in [Wisa]
and is mostly identical to the discussion in [Hag11].
Definition 2.2.8. Let X be a nonpositively curved cube complex. A disc
diagram D → X in X is a continuous combinatorial map of cube complexes,
where D is a disc diagram: a contractible, finite, 2-dimensional cube complex
equipped with a fixed (topological) embedding into S 2 . The area of D is the
number of 2-cubes in D.
Since D is contractible, the complement of D in S 2 is a 2-cell whose attaching map is the boundary path ∂p D of D. If D → X is a disc diagram in X,
then the restriction of this map to the boundary path of D is a combinatorial
path ∂p D → X. Note that ∂p D may not be injective on 0-cubes or 1-cubes.
If X is simply-connected, then any closed combinatorial path in X is the
boundary path of a disc diagram D → X.
Fixing an immersed hyperplane W of X, consider the set of midcubes in
D that map to W . A maximal concatenation of such midcubes is a dual curve
C in D mapping to W . Note that each dual curve is a singular curve: each
1-cube in D has at most two incident 2-cubes, and thus each 0-cell of C has
valence at most 2, though C may cross itself in the interior of one or more
2-cubes.
A 1-cube of D whose midcube is contained in a dual curve C is dual to
C. An end of a dual curve C is a midpoint of a 1-cube of ∂p D dual to C. The
carrier of the dual curve C is the union of closed 2-cubes of D that contain
midcubes belonging to C.
A dual curve C with 0 ends is a nongon. If C is not a nongon, then it
has two ends. A monogon is a closed subpath of a dual curve that crosses
21
itself in the initial and terminal 2-cubes of its carrier. Any dual curve that
crosses itself contains a monogon. An oscugon is a closed subpath C ′ of a dual
curve C such that C ′ does not self-cross, such that the two distinct terminal
1-cubes of c have a common 0-cube but do not form the corner of a 2-cube in
D. A bigon is a pair of dual curves that cross in two distinct squares of D. See
Figure 2–2.
Figure 2–2: Left to right at the top are heuristic pictures of the carriers of
a: nongon, monogon, oscugon, and bigon. Below each of these figures is an
actual disc diagram containing the corresponding configuration. In both sets
of pictures, the dual curve itself is decorated with an arrow.
If C is a dual curve in D whose ends lie on subpaths P, Q of ∂p D, it is
often convenient to say that K emanates from P and terminates on Q (or vice
versa), or that K travels from P to Q.
The techniques used to prove the following lemma are discussed in detail
in [Wisa] and were developed from ideas of Casson.
Lemma 2.2.9 ([Wisa]). Let P → X be a closed combinatorial path in a
nonpositively curved cube complex X and let D → X be a minimal-area disc
diagram among all diagrams D′ with ∂p D′ = P . Then D contains no nongons,
monogons, oscugons, or bigons.
We refer the reader to [Wisa] for a discussion of cancellable pairs and
hexagon moves in disc diagrams over nonpositively-curved cube complexes,
which are used in the proof of Lemma 2.2.9. A detailed discussion of diagrams
with fixed carriers appears in [Hag11] to support the original proof of Theorem 3.1.1. Here, we give a simplified proof of that result that does not use disc
22
diagrams. Although the proof of the projection trichotomy in Chapter 6 does
rely on fixed-carrier diagrams, we shall postpone their mention.
2.2.3
Leaves and cut-compacta
A leaf of X is a hyperplane H such that at least one halfspace H +
associated to H does not contain a hyperplane. This terminology is motivated
by the case in which X is a tree: a leaf of the tree X is a vertex of degree
1, and the midpoint of the single incident edge is a hyperplane of X, one of
whose halfspaces does not contain the midpoint of any edge. The CAT(0)
cube complex X is leafless if no hyperplane H ∈ W is a leaf. Note that, if
X is leafless, then each halfspace contains infinitely many hyperplanes. The
infinite sets of pairwise non-crossing hyperplanes that must exist in a leafless
CAT(0) cube complex X make the property of leaflessness a combinatorial
analogue of the geodesic extension property of geodesic spaces.
Caprace and Sageev discuss essential hyperplanes and essential cube
complexes, and the following definitions come from their paper [CS11]. A
halfspace H + is deep if, for all n ≥ 0, there exists a 0-cube xn ∈ H + such
that dX (xn , N (H)) ≥ n, where H is the hyperplane associated to H + . The
hyperplane H ∈ W is essential if each of its associated halfspaces is deep. The
CAT(0) cube complex X is essential if each of its hyperplanes is essential.
Leaflessness is simply an explicit hypothesis that abstracts the way
essentiality is used in practice. In fact, however, the two notions coincide1 ,
but we prefer to hypothesize leaflessness straightaway.
In Section 2.4.4, we shall discuss leafless (i.e essential ) group actions, and
the essential core theorem of [CS11]. For the moment, we turn our attention
to a small simplification enabled by hypothesizing leaflessness. First, note that
the subcomplex K ⊂ X is compact if and only if W(K) is finite. Indeed, if
K is compact, then K contains finitely many 1-cubes and therefore contains
1
I had originally thought that this requires local finiteness; M. Sageev
pointed out, in personal communication, that the notions are always the same.
23
representatives of finitely many Djoković-Winkler-classes of 1-cubes, i.e.
finitely many hyperplanes cross K. Conversely, if K is crossed by finitely
many hyperplanes, then there are finitely many families of pairwise-crossing
hyperplanes in W(K) and hence finitely many cubes in K.
Definition 2.2.10 (Compactly indecomposable cube complex). Let X
be a CAT(0) cube complex. A cut-compactum in X is a compact, convex
subcomplex K ⊆ X such that X − K has at least two nonempty connected
components. Equivalently, there exist subcomplexes Y1 , Y2 such that each
component of Yi is convex, each Yi properly contains K, and X = Y1 ∪K Y2 .
If X does not contain a cut-compactum, then X is compactly indecomposable.
Definition 2.2.11 (One-ended). The CAT(0) cube complex X is one-ended
if, for each compact subcomplex K ⊂ X, exactly one component of the
closure of X − K is not compact. Equivalently, if X is locally finite, then X is
one-ended if and only if, for each finite subgraph K 1 ⊂ X1 , the complement
X1 − K 1 has exactly one unbounded component.
The following lemma is useful when we consider proper cocompact actions
of a group G on X. The number of ends of a proper metric space is a quasiisometry invariant, and hence G is a one-ended group if and only if X is
one-ended. In the presence of leaflessness, Lemma 2.2.12 allows us to ignore
compact complementary components of a cut-compactum, i.e. to assume that,
if G is one-ended, then X is compactly indecomposable.
Lemma 2.2.12. Let X be locally finite and leafless. Then X is one-ended if
and only if X is compactly indecomposable.
Proof. Suppose X is one-ended and let K ⊂ X be a compact convex subcomplex. Then X − K has a single unbounded component, and a set of compact
components with disjoint interiors, so that
(
)
∪
X = Y ∪K ′
Li ,
i
24
where K ′ ⊂ K, and Y is an unbounded subcomplex containing K ′ and each
Li is a compact subcomplex with Li ∩ Lj ⊆ K ′ for i ̸= j and Li ∩ Y ⊆ K ′ for
all i.
If each Li lies in K ′ , then X = Y, and thus X is compactly indecomposable. If not, then there exists i such that Li contains a 1-cube c dual to a
hyperplane W that does not cross Y and that therefore does not cross Lj for
i ̸= j. In particular, N (W ) is compact and Y lies in a single halfspace W + .
Thus W − is compact, and contains finitely many hyperplanes. X is therefore
not leafless, a contradiction. The converse, that compact indecomposability
implies one-endedness, is obvious from the definitions.
2.2.4
Sageev’s construction, cubes, and the Roller boundary
Sageev’s construction of group actions on CAT(0) cube complexes is
based on the following notions. Let G be a finitely-generated group and let
H ≤ G. Then H is a codimension-1 subgroup if there exists r < ∞ such
that the closed r-neighborhood Nr (H) of H in (some fixed locally finite
Cayley graph of) G has the property that G − Nr (H) has at least two deep
components. The component C of G − Nr (H) is deep if for all n ≥ 0,
there exists c ∈ C such that dG (c, Nr (H)) ≥ n. Note that, by grouping
the complementary components of Nr (H) into two sets in some way, and
arbitrarily adding Nr (H) to one of these sets, each codimension-1 subgroup
H ≤ G, and each left coset of H, induces a bipartition of G. On the other
hand, if X is a CAT(0) cube complex, then each hyperplane H induces a
bipartition of X0 : the sets in the bipartition are the sets of 0-cubes in the
halfspaces associated to H.
These two examples are wallspaces in the sense of [HP98]. Following
Chatterji-Niblo [CN05], we give an account of Sageev’s construction in the
more general context of wallspaces; Sageev’s original construction yields
a G-cube complex from a codimension-1 subgroup [Sag95], but the more
general construction is almost identical. We have chosen the viewpoint
in [CN05], emphasizing contacting walls, because it simplifies our discussion
25
somewhat; we could equally well have used the language of median algebras
and ultrafilters adopted by Nica in [Nic04].
Definition 2.2.13 (Wallspace). A wallspace is a nonempty set S – the
underlying set – together with a nonempty collection W of walls. A wall
W ∈ W is a partition S = W + ⊔ W − into two nonempty subsets called
halfspaces. If s, s′ ∈ S, we say that the wall W separates s from s′ if s, s′
lie in distinct halfspaces associated to the wall W. The strong finite interval
condition requires that for all s, s′ ∈ S, finitely many walls separate s from s′ .
We require wallspaces to satisfy the strong finite interval condition.
The wallspace (S, W) is Hausdorff if any two elements of S are separated by at least one wall; the wallspaces arising from hyperplanes in cube
complexes are Hausdorff. If (S, W) is a Hausdorff wallspace, then for each
s, s′ ∈ S, the function
♯(s, s′ ) = |{W ∈ W : W separates s, s′ }|
defines a metric on S. If (S, W) is not Hausdorff, then ♯ is a pseudometric.
If W1 , W2 are walls such that all four quarterspaces W1+ ∩ W2+ , W1+ ∩
W2− , W1− ∩ W2+ , W1− ∩ W2− are nonempty, then W1 , W2 cross. If W1 , W2 , W3
are walls, then W2 separates W1 from W3 if W1− ⊂ W2− and W3+ ⊂ W2+
(or any of the other analogous pairs of containments arising from relabeling
halfspaces). If W1 , W2 are not separated by a wall W3 , then they contact. Note
that crossing walls necessarily contact.
Construction 1 (Cubulating wallspaces). Let (S, W) be a wallspace and let
W ± be the set of halfspaces. Let π : W ± → W be defined by π(W ± ) = W,
i.e. π is the map that sends each halfspace the associated wall. The dual cube
complex X = X(S, W) is constructed as follows.
First, a 0-cube is a section x : W → W ± of π. This is often referred to as
an orientation of all walls, i.e. x chooses a halfspace associated to each wall. x
must satisfy an additional consistency condition, which says that for all walls
W, W ′ ∈ W, the intersection x(W ) ∩ x(W ′ ) of the halfspaces designated by x
26
is nonempty – x does not orient two walls away from one another. Note that,
if W, W ′ cross, then any section of π restricts to a consistent orientation of the
pair W, W ′ . For this reason, crossing walls are sometimes called independent.
Note that for each s ∈ S, there is an associated 0-cube xs : for each
W ∈ W, let xs (W ) be the unique halfspace associated to W that contains s. A
0-cube x is canonical if there exists s ∈ S such that x = xs . Let X0 be the set
of canonical 0-cubes, together with any 0-cube x that differs on finitely many
walls from a canonical 0-cube.
Figure 2–3: The three pictures at the top show orientations of non-crossing
walls: the third is inconsistent. At the bottom: crossing walls can be oriented
independently, and the four possible orientations determine the 1-skeleton of a
2-cube.
If x, y are 0-cubes (not necessarily canonical), then x, y are joined by a
1-cube if and only if there is a unique wall W for which x(W ) ̸= y(W ). Note
that, by the strong finite interval condition, if xs , xt are canonical 0-cubes
associated to s, t ∈ S respectively, then xs (W ) ̸= xt (W ) for finitely many
walls W. Indeed, xs (W ) ̸= xt (W ) if and only if W separates s, t, by definition,
but the strong finite interval condition guarantees that there are finitely many
such W. We therefore let X1 be the graph whose 0-skeleton is X0 , and whose
1-cells are the 1-cubes that join 0-cubes of X0 . The above argument shows
that X1 is path-connected. Moreover, if xs , xt are canonical 0-cubes, the
27
standard path-metric dX on X1 is given by:
dX (xs , xt ) = ♯(s, t).
Indeed, if s, t are not separated by any walls, then xs = xt by definition.
One then checks that a shortest path joining the canonical 0-cubes xs , xt is
obtained by successively “flipping” the finitely many walls separating xs and
xt , in such a way that each reversal of orientation of a single wall yields a new
0-cube.
One now verifies (see e.g. [CN05] or [Nic04]) that X1 is the 1-skeleton of a
uniquely-determined CAT(0) cube complex X, which is the cube complex dual
to (S, W).
Mostly, we shall work with wallspaces that arise from pre-existing CAT(0)
cube complexes and hyperplanes. Accordingly, we emphasize the following
viewpoint on cubes. As always, X is a CAT(0) cube complex, W is the set of
hyperplanes, W ± is the set of halfspaces, and π : W ± → W is the two-to-one
map that sends each halfspace to its corresponding hyperplane. Let x0 ∈ X0
be a fixed base 0-cube. Then a 0-cube x ∈ X is a map x : W → W ± with the
following properties:
1. (Section of π.) For each hyperplane H ∈ W, we have π(x(H)) = H.
2. (Consistency.) For each H, H ′ ∈ W, we have x(H) ∩ x(H ′ ) ̸= ∅.
3. (Canonical cube.) The set of hyperplanes H ∈ W for which x(H) ̸=
x0 (H) is finite.
Condition (3) is independent of the choice of x0 . The 0-cubes x, y are
joined by a 1-cube if and only if there exists a single hyperplane H for which
x(H) ̸= y(H). More generally, for x, y ∈ X0 , the distance dX (x, y) in
the graph X1 counts the number of hyperplanes H on which the maps x, y
differ. The 2n 0-cubes x1 , . . . , x2n form the 0-skeleton of an n-cube if and
only if there is a collection H1 , . . . , Hn of pairwise-crossing hyperplanes such
that, for 1 ≤ i < j ≤ 2n , if H is a hyperplane with xi (H) ̸= xj (H), then
H = Hk for some k ≤ n. This viewpoint is of particular utility when we
28
discuss subcomplexes and quotients of X. If A ⊂ X is a subcomplex, H is
a hyperplane such that A ⊂ H + , and x is a 0-cube, we say that x orients
H toward A if H + = x(H). Note that each hyperplane H induces a wall on
X0 : the sets in the bipartition are H ± ∩ X0 . One easily checks that the cube
complex dual to the wallspace whose underlying set is X0 and whose walls are
induced by the hyperplanes is exactly X.
What happened to the non-canonical 0-cubes? A 0-cube at infinity is a
section f of π that satisfies the consistency condition but that does not satisfy
the canonical cube condition. In other words, for some (and thus any) 0-cube
x0 , the maps f and x0 differ on infinitely many hyperplanes. The Roller
boundary of X, introduced in [Rol98], captures these cubes at infinity; our
description of the Roller boundary follows [NS11], although we have translated
this from the language of ultrafilters for consistency with the above discussion.
Let R denote the set of all consistent sections f : W → W ± of π. There
is a natural inclusion X0 ,→ R, since X0 consists of those f ∈ R satisfying
the additional condition (3). The set ∂R X = R − X0 is the set of 0-cubes at
infinity. ∂R X is topologized as follows: for each halfspace H + , let UH + be the
set of all x ∈ R such that x(H) = H + . Take {US : S ∈ W ± } as a sub-basis
of a topology T on R; the subspace ∂R X, with the induced topology, is the
Roller boundary.
Note that this topology coincides with the metric topology on X0 coming
from the path-metric dX on the 1-skeleton. Indeed, each US ∩ X is the set
of 0-cubes in the halfspace S (associated to some hyperplane H), which is
open in the metric topology. Conversely, X0 with the metric topology is a
discrete space, and thus US is open. Roller showed in [Rol98] that X0 is open
∏
and dense in R. Note that R injects into the product H∈W {H − , H + }, and
that the topology T coincides with the topology induced by the Tychonoff
∏
topology on the compact space H∈W {H − , H + } (see [NS11]). In particular,
R is compact, and hence ∂R X, being the complement in a compact space
of an open set, is compact. Although we will not use the Roller boundary,
29
this fact is important because it distinguishes the Roller boundary from the
cosmetically similar simplicial boundary introduced in Chapter 5.
2.2.5
Restriction quotients
Given a subset of W, one can define a cubical quotient of X; intuitively,
the hyperplanes of X that are not contained in the designated subset are
“collapsed”. The resulting quotient complex is the restriction quotient
discussed in [CS11].
Let V ⊆ W be a set of hyperplanes of X, and let V be the set of walls
in the set X0 induced by the hyperplanes in V: for each hyperplane V ∈ V,
the induced wall is the partition (V + ∩ X0 ) ⊔ (V − ∩ X0 ), where V ± are the
halfspaces of X associated to V. Let V± = V ± ∩ X0 be the halfspaces of the
wall in X0 associated to V.
Let X(V) be the cube complex dual to the wallspace
(
)
X0 , {V± : V ∈ V} .
We define a cubical map q : X → X(V) as follows. For each 0-cube x : W →
W ± , let q(x) = x|V . This restriction of x to V is automatically a consistent,
canonical section of π|V , and hence a 0-cube of X(V), by Construction 1. Note
that q(x) = q(y) if and only if x, y only differ on hyperplanes in W − V.
If x, y are adjacent 0-cubes, differing on a single hyperplane H, then
either H ̸∈ V, or H is the unique element of V on which q(x), q(y) differ.
Hence q(x) and q(y) either coincide or are adjacent. It is now routine to check
that q extends to a cubical quotient q : X → X(V). This construction plays
an important role in the characterization of bounded contact graphs (see
Chapter 6) and in the essential core theorem of Caprace-Sageev, among other
places.
Two points are worth noting. First, if H, H ′ ∈ W cross, then q(H)⊥q(H ′ ).
Conversely, if q(H)⊥q(H ′ ), then H, H ′ ∈ V cross in X. Similarly, if H, H ′ osculate, then their images in X osculate. However, it is possible that H, H ′ do
not contact, but every hyperplane H ′′ separating H from H ′ lies in W − V. In
30
such a case, q(H) and q(H ′ ) osculate in X. In particular, dim X(V) ≤ dim X,
but no analogous inequality governs the degree of X(V). In fact, we shall see
an example of a proper, cocompact CAT(0) cube complex and an equivariant
restriction quotient that is not locally finite.
2.3
Contact and crossing graphs
The crossing relation on W is encoded by the transversality graph
introduced by Roller [Rol98] and independently, as the crossing graph,
in [Hag11]. Sageev’s construction exposes some difficulties with the crossing
graph, and we therefore introduce the contact graph, which is probably the
main tool in this work. This discussion follows that in [Hag11], except where
noted otherwise.
Definition 2.3.1 (Crossing graph, contact graph). The contact graph ΓX
of the CAT(0) cube complex X is the (simplicial) graph whose vertex-set is
the set W of all hyperplanes, with H, H ′ ∈ W adjacent if H ⌣
⊥ H ′ . The
crossing graph ∆X is the subgraph of ΓX whose vertex set is again W, with
H, H ′ ∈ W adjacent if H⊥H ′ . The closed edge in ΓX joining the contacting
hyperplanes H, H ′ is also denoted H ⌣
⊥ H ′ , or H⊥H ′ if the corresponding
hyperplanes cross and we wish to emphasize this fact. An edge of ΓX of the
form H⊥H ′ is a crossing-edge; otherwise, H ⌣H
⊥ ′ is an osculation-edge.
Several simple properties of X are encoded by ΓX. For example, each dcube in X is dual to a collection of d pairwise-crossing hyperplanes, and hence
dim X is equal to the cardinality of a largest clique in ∆X. More generally, if
x ∈ X is a 0-cube of valence d, then there are d hyperplanes that pairwisecontact at x, and hence the cardinality of a largest clique in ΓX is exactly the
valence of a highest-valence 0-cube in X. The CAT(0) cube complex X is thus
locally finite if every clique in ΓX has finite cardinality, and finite-dimensional
if the cliques in ∆X have uniformly bounded cardinality.
Example 2.3.2 (Contact graph examples). Figure 2–1 shows a simple
example of a CAT(0) cube complex, its crossing graph, and its contact graph.
The crossing graph and contact graph of a d-cube are both isomorphic to a
31
complete graph on d vertices, since the hyperplanes of a cube pairwise-cross.
The crossing graph of a tree T is a totally disconnected graph with one vertex
for each edge of T, since hyperplanes are midpoints of edges of T and no two
hyperplanes cross (otherwise there would be a 2-cube). The contact graph
of T has an edge joining the hyperplanes V, W if and only if the edges of T
whose midpoints are V, W have a common vertex; in other words, ΓT is the
graph dual to T.
If Y is an isometrically embedded subcomplex, there is a subgraph
Λ ⊆ ΓX whose vertices are the hyperplanes H that cross Y , with H, H ′
joined by an edge in Λ if N (H) ∩ N (H ′ ) ∩ Y ̸= ∅. Note that an isometrically
embedded subcomplex need not itself be CAT(0), so that the graph Λ is not
necessarily the contact graph of a CAT(0) cube complex. On the other hand,
if Y is convex, then the inclusion Λ ,→ ΓX is a graph monomorphism that
sends crossing-edges to crossing-edges and osculation-edges to osculationedges. By Proposition 2.2.6, Λ is a full subgraph of ΓX. Conversely, if Y is an
isometrically embedded CAT(0) cube complex, then Λ is the contact graph of
Y and, if Λ is a full subgraph of ΓX, then Y is convex.
An important special case is that of an isometrically embedded interval,
i.e. a geodesic segment, ray, or bi-infinite geodesic. Let I be an interval (either
finite, R+ , or R) and let γ : I → X be an isometric embedding. Let Hi be the
hyperplane dual to the ith 1-cube of I. More precisely, I = [− 12 + a, − 21 + b]
for a, b ∈ Z ∪ {±∞}, and Hi is the hyperplane of X dual to the 1-cube
⊥ i+1 , since N (Hi ) ∩ N (Hi+1 )
γ([− 12 + i, 12 + i]). Then for each i, we have Hi ⌣H
contains the 0-cube common to the ith and (i + 1)th 1-cubes of γ. The
projection of γ to ΓX is the path
ProjΓ γ = . . . H0 ⌣H
⊥ 1⌣
⊥ ...
in γ. We use the notation Λ(γ) for the full subgraph of ΓX induced by
ProjΓ γ.
32
Note, however, that full subgraphs of ΓX do not in general correspond to
convex subcomplexes of X. This issue is addressed in the following definition
and propositions:
Definition 2.3.3 (Inseparable set). The subset V ⊆ W is inseparable if, for
each V, V ′ ∈ V and for any hyperplane V ′′ that separates V from V ′ , we have
V ′′ ∈ V. Note that, if Y ⊆ X is a connected subcomplex, then the set of
hyperplanes crossing Y is inseparable.
The first proposition says that finite inseparable sets of hyperplanes
induce convex subcomplexes. This statement follows from part of the proof of
Lemma 7.9 in [Hag11], but we make it explicit here.
Proposition 2.3.4. Let V ⊆ W be a finite, inseparable set of hyperplanes,
and let Γ be the full subgraph of ΓX induced by the vertices corresponding to
hyperplanes in V. Then there exists a convex subcomplex Y ⊆ X such that
W(Y) = V and ΓY ∼
= Γ.
Proof. Denote by V the set of walls in X0 induced by the hyperplanes in V
and by Y the CAT(0) cube complex dual to the wallspace (X0 , V). Fix a base
0-cube x0 ∈ N (W0 ), where W0 ∈ V is a hyperplane such that all hyperplanes
in V either cross W0 or lie in a fixed halfspace W0+ . Such a W0 exists because
V is finite. Choose x0 so that each V ∈ V with V ⊂ W0+ is separated from x0
by W0 .
Each 0-cube y ∈ Y is a consistent, canonical map y : V → V±
that chooses a halfspace of X0 for each hyperplane in V. Define a map
ψ : Y0 → X0 as follows. For each W ∈ V, let ψ(y)(W ) be the halfspace
associated to W that contains the halfspace y(W) ⊂ X0 , where W ∈ V is the
wall associated to W. For each of the finitely many W ∈ W − V, such that W
crosses W0 and separates some V ∈ V from x0 , let ψ(y)(W ) = X − x0 (W ).
Otherwise, let ψ(y)(W ) = x0 (W ).
The map ψ(y) is a section of π : W ± → W, by definition. Moreover,
ψ(y) is canonical: since V is finite, ψ(y) differs from x0 on finitely many
33
hyperplanes. Inseparability guarantees the map ψ(y) consistently orients
the hyperplanes of X. Indeed, if W, W ′ ∈ V, then ψ(y)(W ) ∩ ψ(y)(W ′ ) is
nonempty, since it contains each 0-cube of X contained in y(W) ∩ y(W′ ),
which is nonempty since y consistently orients the walls in V. If W, W ′ ̸∈ V,
then ψ(y)(W ) = x0 (W ) and ψ(y)(W ′ ) = x0 (W ′ ), so that their intersection
is nonempty because x0 consistently orients the hyperplanes of X. Finally,
suppose that W ∈ V and W ′ ∈ W − V. If W ⊥W ′ , then the corresponding
walls are independent and ψ(y) automatically orients W, W ′ consistently. If
not, then there are three possibilities. If neither of W and W ′ separates the
other from x0 , then W lies in x0 (W ′ ) and either possible value of ψ(y)(W )
leads to a consistent orientation. If W separates W ′ from x0 , then ψ(y)(W )
must intersect ψ(y)(W ′ ) = x0 (W ) regardless of the choice of ψ(y)(W ).
The only remaining situation is that in which W ′ separates W from x0 .
In this case, W ′ must cross W0 , for otherwise W ′ would separate W0 from
W0 , contradicting the fact that V is inseparable. Thus ψ(y)(W ′ ) orients W ′
away from x0 , by definition, and thus toward W . Either choice of halfspace
for ψ(y)(W ) is consistent with ψ(y)(W ′ ). Hence ψ : Y0 → X0 is an isometric
embedding. Indeed, ψ(y), ψ(y ′ ) differ exactly on the hyperplanes in V corresponding to walls in V on which y, y ′ differ. We thus obtain an isometric
embedding Y → X.
Since the hyperplanes in X crossing (the image of) Y correspond bijectively to the walls of (X0 , V), and these walls are precisely those induced by
V, we have W(Y) = V. On the other hand, two walls in V contact if and only
if the corresponding hyperplanes in V contact in X, so that ΓY is isomorphic
to the full subgraph of ΓX induced by V, i.e. to Γ. By Proposition 2.2.6, Y is
convex in X.
We shall see an example in Chapter 5 of an infinite inseparable set of
hyperplanes that does not induce a subcomplex in this way; this is the crux of
the notion of visible points at infinity discussed in that chapter. The problem
is roughly that Y may embed as a collection of cubes at infinity. However,
34
at least in one important special case, subgraphs induce subcomplexes.
The following is an application of Sageev’s construction, proved in detail
in [Hag11]. On certain occasions where we could apply Proposition 2.3.5, we
instead proceed explicitly.
Proposition 2.3.5. Let Y, X be locally finite CAT(0) cube complexes,
with contact graphs ΓY, ΓX and crossing graphs ∆Y, ∆X respectively. Let
ϕ : (ΓY, ∆Y) → (ΓX, ∆X) be an injective graph homomorphism that is
surjective on vertices. Suppose, moreover, that if U, V, W are hyperplanes of Y
such that V separates U and W , then either ϕ(V ) separates ϕ(U ) from ϕ(W ),
or ϕ(V ) crosses one of the hyperplanes ϕ(U ) or ϕ(W ). Then there is a cubical
isometric embedding ψ : Y → X. In particular, X is uniquely determined, up
to cubical isomorphism, by the pair (ΓX, ∆X).
The next proposition was first proved by Roller in [Rol98] and independently by the author in [Hag11]. In the guise of intersection graphs of convex
splits in median graphs, this statement, and the method of proof in both of
the aforementioned papers, seems to be widely known.
Proposition 2.3.6. Let ∆ be a connected simplicial graph. Then there exists
a CAT(0) cube complex X such that ∆X ∼
= ∆.
The fact that every simplicial graph is a crossing graph means that there
is little hope of any general geometric statements about crossing graphs of
cube complexes. Also, crossing graphs can be disconnected, while it is obvious
from projection to the contact graph that ΓX is always connected.
Moreover, ∆ alone does not uniquely determine X. For example, the
graph K2,3 is the crossing graph of the product I2 × I3 , where Ij is an interval
of length j, but K2,3 is also the crossing graph of I2 × T, where T is a tripod.
However, the property of being a cubical product is detected at the level of
the crossing graph.
2.3.1
Cubical products
Let X1 , X2 be CAT(0) cube complexes, and let Wi be the set of hyperplanes in Xi . Consider the cube complex X ∼
= X1 × X2 . The 0-skeleton of
35
X can be viewed as the set of pairs (x1 , x2 ) of consistent, canonical sections
xi : Wi → Wi± of the projection πi : Wi± → Wi . The 0-cubes (x1 , x2 ) and
(x′1 , x′2 ) are adjacent if x1 = x′1 and x2 , x′2 differ on a single hyperplane, or
vice-versa. It is not hard to verify the link condition; it is even less hard to see
that X is simply connected. Hence X is a CAT(0) cube complex and therefore
contains hyperplanes.
Let s be a 2-cube of X, with 0-cubes (xi1 , xi2 ), where i ∈ {1, 2} and x1j , x2j
are adjacent 0-cubes of Xj , for j ∈ {1, 2}. Then (x11 , x12 ) is adjacent to (x21 , x12 ),
and the resulting 1-cube is parallel to that joining (x11 , x22 ) to (x21 , x22 ). Taking
the transitive closure of the parallelism relation shows that W(X) = W 1 ⊔ W 2 ,
where
W 1 = {h = H × X2 : H ∈ W1 }
W 2 = {v = X1 × V : V ∈ W2 },
and that, if c ⊂ Xi is a 1-cube dual to a hyperplane H, then its image in X is
dual to the corresponding hyperplane of X. If h ∈ W 1 and v ∈ W 2 correspond
to H ∈ W1 , V ∈ W2 respectively, h⊥v. Indeed, if c ⊂ X1 is a 1-cube dual to
H, and c′ ⊂ X2 a 1-cube dual to V , then c × c′ is a 2-cube of X, containing
the images of c and c′ . Hence h and v cross in c × c′ . Since Xi ⊆ X, the
hyperplanes H, H ′ of Xi cross (osculate) if and only if their images cross
(osculate) in X. This yields:
Proposition 2.3.7. Let X ∼
= X1 × X2 . Then ∆X ∼
= ∆X1 ⋆ ∆X2 . Furthermore,
ΓX ∼
= ΓX1 ⋆ ΓX2 , and every edge joining a vertex in ΓX1 to a vertex in ΓX2
is a crossing edge.
Proposition 2.3.7 and Proposition 2.3.8 are translations into contact
graph language of well-known facts that appear, for example, in [CS11].
Proposition 2.3.8. Let X be a CAT(0) cube complex containing convex
subcomplexes X1 , X2 such that, for all H ∈ W(X1 ), V ∈ W(X2 ), the
36
hyperplanes H and V cross. Then there is a cubical isometric embedding
X1 × X2 → X. In particular, if W = W1 ⊔ W2 , and every H ∈ W1 crosses
every V ∈ W2 , then X decomposes as the product of the convex subcomplexes
crossed by the sets W1 , W2 .
Proof. Let (x1 , x2 ) ∈ X01 × X20 . Choose H0 ∈ W1 = W(X1 ), V0 ∈ W2 = W(X2 ),
and a 0-cube x ∈ N (H0 ) ∩ N (V0 ). Let ϕ(x1 , x2 ) : W → W ± be a section
of π defined as follows. For H ∈ W1 , let ϕ(x1 , x2 )(H) be the halfspace of X
associated to H that contains x1 , and define ϕ(x1 , x2 )(V ) in the analogous
way, using x2 , when V ∈ V2 . Since xi consistently orients the hyperplanes
in Wi , this defines consistent orientations of W1 and of W2 . Moreover, the
orientation of H ∈ W1 , V ∈ W2 provided by ϕ(x1 , x2 ) is automatically
consistent since H⊥V. The orientation ϕ(x1 , x2 ) differs from x on exactly the
finitely many hyperplanes on which either x1 or x2 differs from x. Hence, to
show that ϕ(x1 , x2 ) yields a 0-cube of X, it suffices to extend this map to an
orientation of hyperplanes W ∈ W − (W1 ∪ W2 ) in a consistent, canonical
fashion.
For each hyperplane W ̸∈ W1 ∪ W2 , that does not separate X1 from
X2 , let ϕ(x1 , x2 )(W ) = x(W ). This is consistent across all hyperplanes in
W − (W1 ∪ W2 ), since x is a 0-cube of X and therefore orients all hyperplanes
in X consistently. Moreover, if H ∈ W1 ∪ W2 and W ̸∈ W1 ∪ W2 , then either
H⊥W , in which case ϕ(x1 , x2 ) automatically orients H and W consistently, or
H and W do not cross. Now W does not separate H from H0 , since otherwise
W would cross X1 and hence lie in W1 . If W separates H from x0 , then
W must therefore cross H0 , and be among the finitely many hyperplanes
separating X1 from X2 , since W cannot cross X2 , a contradiction. Hence x
orients W toward H, and therefore ϕ(x1 , x2 ) orients H, W consistently and
differs from x on finitely many hyperplanes.
If W separates X1 from X2 , then for each V ∈ W2 , the subcomplex
N (V ) ∩ X2 is separated from X1 by W . If W does not cross V , then W
must cross every H ∈ W1 . Hence the finitely many hyperplanes W that
37
separate X1 from X2 have the property that W ⊥H for all H ∈ V1 or W ⊥V
for all V ∈ W2 . If W is of the former type, let ϕ(x1 , x2 )(W ) be the halfspace
containing X2 . Otherwise, let ϕ(x1 , x2 )(W ) be the halfspace containing X1 .
This orientation is consistent, and hence ϕ(x1 , x2 ) is a consistent section of
π. Moreover, ϕ(x1 , x2 ) differs from the canonical 0-cube x exactly on finitely
many hyperplanes, and hence ϕ(x1 , x2 ) ∈ X0 . Finally, if x1 , x′1 ∈ X01 , x2 , x′2 ∈
X02 , then ϕ(x1 , x2 )(W ) ̸= ϕ(x′1 , x′2 )(W ) exactly for those W for which either
x1 (W ) ̸= x′1 (W ) or x2 (W ) ̸= x′2 (W ), i.e.
dX (ϕ(x1 , x2 ), ϕ(x′1 , x′2 )) = dX1 (x1 , x′1 ) + dX2 (x2 , x′2 ).
Hence there is a cubical isometric embedding ϕ : X1 × X2 → X.
In particular, if every hyperplane belongs to W1 ⊔ W2 , then the above
construction shows that X is isomorphic to the product of X1 and X2 .
Cubical products and their generalizations, pseudo-products, play an
important role in Chapter 6, where we characterize CAT(0) cube complexes
with bounded contact graphs.
2.3.2
Recubulation
Via recubulation, contact graphs can be canonically converted into
crossing graphs. Recubulation is a local construction, and for added generality,
we work with a nonpositively-curved (not necessarily simply-connected) cube
complex X, with CAT(0) universal cover X → X. An immersed hyperplane
in X is the image of an immersion H̄ → X, where H̄ is the image of a
hyperplane H ⊂ X under the universal covering projection.
Proposition 2.3.9 (Recubulation). Let X be a nonpositively-curved
cube complex with a set W of immersed hyperplanes. Then there exists a
nonpositively-curved cube complex Xr , with set Wr of immersed hyperplanes,
such that X ⊆ Xr and the inclusion X ,→ Xr induces a bijection b : W → Wr
such that, for all V, W ∈ W, the immersed hyperplanes b(V ) and b(W ) cross if
and only if V ⌣W
⊥ . Moreover, we have:
38
1. The dimension of Xr is equal to the degree of X.
2. Xr deformation retracts to X.
3. Xr is compact if and only if X is compact.
Proof. The complex Xr is constructed as follows. Let V, W be a pair of
immersed hyperplanes in X and let c be a 0-cube such that V and W osculate
at c, i.e. there exists a 1-cube v dual to V and a 1-cube w dual to W such
that v and w both contain c and do not form the corner of a 2-cube. For each
such triple c, v, w determine by V, W , attach a 2-cube s to X by identifying
two consecutive 1-cubes of s with vw in the obvious way. The immersed
hyperplanes V, W extend into s in such a way that the extensions cross. The
′
complex Xr , consisting of X, together with the new 2-cubes may fail to be
nonpositively curved. To rectify this, we add higher-dimensional cubes, so
′
that Xr is the minimal nonpositively-curved cube complex containing Xr .
By construction, each immersed hyperplane of X extends to an immersed
hyperplane of Xr in such a way that immersed hyperplanes of Xr cross if and
only if their intersections with X contact.
Proof of (1): This is immediate, since the dimension of Xr is the
largest cardinality of a set of pairwise-crossing immersed hyperplanes, which
corresponds to a set of pairwise-contacting immersed hyperplanes in X of the
same cardinality.
Proof of (2): The deformation retraction is constructed by collapsing
along free faces. Each new 2-cube of Xr has exactly two 1-cubes that do not
lie in X.
Proof of (3): Since X ⊆ Xr is a subcomplex, one direction is obvious. If
X is compact, then there are finitely many osculations, and hence Xr − X has
finitely many 2-cubes. Thus Xr has a finite 2-skeleton and therefore contains
finitely many corners of missing 3-cubes. By induction, Xr contains finitely
many d-cubes for each d ≥ 0. For each d-cube c of Xr , there is a family of d
pairwise-contacting (not necessarily distinct) immersed hyperplanes V1 , . . . , Vd
of X such that there is a 0-cube c of X that is contained in distinct 1-cubes
39
v1 , . . . vd with vi dual to Vi for each i. Since X is compact, there is a bound on
d, and thus Xr is a finite-dimensional cube complex with finitely many cubes
of each dimension.
Intuitively, recubulation turns all osculations in X into crossings (and
therefore may introduce new osculations); see Figure 2–4. Note that, if X is
CAT(0), then Xr is also CAT(0) and ∆Xr ∼
= ΓX.
Figure 2–4: Turning an osculation into a crossing.
The main extant use of recubulation is in [CH11]. Beginning with a
certain 4-dimensional CAT(0) cube complex built, via Sageev’s construction,
from the box complex of Burling [Bur65], we used recubulation to construct
a 5-dimensional, finite-degree CAT(0) cube complex that does not embed in
the product of finitely many trees. Below, we recubulate to simplify the proof
that cocompactness of a proper action on a cube complex corresponds to a
cocompact action on the contact graph, although recubulation is probably not
conceptually essential.
2.4
Groups acting on cube complexes
In this section, we describe the type of group action we shall study, and
give a few preparatory results. In Section 2.4.6, we state Sageev’s fundamental
theorem about groups acting on cube complexes, and discuss some examples
of such groups.
2.4.1
Actions on cube complexes and contact graphs
G always denotes a countable discrete group. By an action of G on X,
we mean a monomorphism G → Aut(X) of G into the group of cubical
automorphisms X → X. Such an action restricts to an action by isometries
on (X1 , dX ). As a result, if H ∈ W is a hyperplane, then for all g ∈ G,
the subspace gH is also a hyperplane. Indeed, if c, c′ are parallel 1-cubes,
40
then gc, gc′ are also parallel because G acts isometrically on X1 . Hence G
preserves the Djoković-Winkler relation on the set of 1-cubes, and therefore
acts on W. Moreover, if H⊥H ′ , then there is a 2-cube s ⊂ N (H) ∩ N (H ′ )
whose 1-cubes are dual to H, H ′ . Hence for each g ∈ G, the 2-cube gs is
dual to gH, gH ′ , and those hyperplanes therefore cross. The same argument
holds when H, H ′ osculate, with s replaced by a pair c, c′ of 1-cubes, dual to
H, H ′ respectively, with a common 0-cube. Hence G acts on ΓX, and ∆X is a
G-invariant subgraph.
Note that G may act on ΓX with edge-inversions, since there may be g ∈
G and two hyperplanes H1 , H2 such that H1 ⌣
⊥ H2 and gH1 = H2 , gH2 = H1 .
Hence we will not speak of the quotient G\ΓX. Instead, we say that G acts
cocompactly on ΓX if there are finitely many G-orbits of vertices and edges.
In Section 2.4.3, we show that, if G acts properly on X, then there are finitely
many orbits of hyperplane contacts if and only if G acts cocompactly on X.
2.4.2
Proper actions
If A ⊆ X is a subcomplex, or Λ ⊆ ΓX a subgraph, we denote by GA
and GΛ their respective stabilizers. We say that G acts properly on X if the
stabilizer of each cube is finite. If X is locally finite – i.e. if X1 is a proper
metric space – then G acts properly on X if and only if G acts metrically
properly on X1 .
For d ≥ 1, let c be a d-cube of X that is maximal, i.e. not properly
contained in any cube. Let c be dual to the set H1 , . . . , Hd of pairwise-crossing
hyperplanes and let Λ(c) be the d-clique in ΓX generated by these hyperplanes. Note that, by maximality of c, there is a unique convex subcomplex
of X, namely c, whose contact graph is Λ(c). If g ∈ Gc , then g ∈ GΛ(c) , since
g must stabilize the set of 1-cubes of c and thus the set of Djoković-Winkler
classes of 1-cubes represented by the 1-cubes of c. On the other hand, if
g ∈ GΛ(c) , then for all i, gHi = Hj for some j. In particular, gc is a d-cube
dual to H1 , . . . , Hd and thus gc = c since c is the unique cube dual to that
collection of hyperplanes. Hence Gc = GΛ(c) . If c′ ⊂ c is a face, then Gc′
41
stabilizes the union of the maximal cubes containing c′ . Thus, if X is locally
finite, then G acts properly on X if and only if the stabilizer of each maximal
clique in ∆X is finite.
Following Wise, we say that G is cubulated if there exists a CAT(0) cube
complex X such that G acts properly on X; the action is termed a cubulation.
Having constructed a G-action on a CAT(0) cube complex X, there are
several ways of verifying properness, depending on the context.
For example, if G acts on a wallspace (S, W), then G acts properly on the
dual cube complex X provided that, for each s ∈ S, we have ♯(s, gs) → ∞ as
g → ∞. In particular, this holds when S has some metric d such that G acts
isometrically on (S, d), and the function defined by
S 2 ∋ (s1 , s2 ) 7→ ♯(s1 , s2 ) ∈ R
is bounded below by a linear function of d(s1 , s2 ). This is the linear separation
property from [HWb] (see [Wis11]). Other methods of verifying that an action
on a cube complex is a cubulation include the cut-wall criterion [Wis11], but
we omit detailed explanation of such criteria since we are generally concerned
with preexisting proper actions on cube complexes, rather than with creating
cubulations.
2.4.3
Cocompact actions and the contact graph
There are criteria guaranteeing that the action of G on X (or, equivalently, on the median graph X1 ) is cocompact. The first of these, due to
Sageev, says that if G is a word-hyperbolic group that acts on a wallspace
(S, W) with finitely many orbits of walls, and the stabilizer of each wall is
a quasiconvex subgroup of G, then G acts cocompactly on the dual cube
complex [Sag97]. Hruska-Wise have generalized this to certain relatively
hyperbolic groups in [HW10], by defining cosparse actions.
Note that if G acts cocompactly on X, then X is finite-dimensional. If G
acts properly and cocompactly, then X is finite-dimensional and, moreover,
uniformly locally finite. In other words, there is a uniform bound on the
42
size of cliques in ΓX. The result of this section, Proposition 2.4.2 says that
cocompactness is visible on the level of the contact graph. We need a simple
lemma about recubulations.
Lemma 2.4.1. Let G act properly and cocompactly on the CAT(0) cube
complex X and let Xr be the recubulation of X. Then G acts properly and
cocompactly on Xr and X is a G-invariant subcomplex.
Proof. We first define the action of G on Xr as follows, extending the action
of G on X. For each 0-cube c whose interior lies in Xr − X, there is a corner
k(c) of c in X, by the definition of recubulation. The subcomplex K(c) is the
union of those faces of c that lie in X. Then for each g ∈ G, the subcomplex
gK(c) also forms the corner of a cube c′ whose interior lies in Xr − X, and
we let gc = c′ . It is easily seen that this defines an action making X a Ginvariant subcomplex. Hence the stabilizer of c stabilizes K(c), and thus G
acts properly on Xr , since K(c) is a finite subcomplex and G acts properly on
X. Let X = G\X. Then, by definition, (X)r = Xr , i.e. the recubulation of
the quotient is equal to the quotient of the recubulation by the G-action. By
Proposition 2.3.9.(3), Xr is compact.
Proposition 2.4.2. Let G act properly on the CAT(0) cube complex X. This
action is cocompact if and only if the induced action on ΓX is cocompact.
Proof. Suppose that G acts properly and cocompactly on X. Then, by
cocompactness, there are finitely many orbits of hyperplanes, so that G\ΓX0
is finite. Let H1 , . . . , Hk be a complete set of G-orbit representatives of
∪
hyperplanes. Then ∆X = G∆k , where ∆k = ki=1 B1 (Hi ) and B1 (Hi ) is the
closed 1-ball in ∆X centered at Hi . Indeed, each edge of ∆X is of the form
gHi ⊥hHj for some g, h ∈ G and some 1 ≤ i, j ≤ k. This edge lies in gB1 (Hi ).
Since G acts properly and cocompactly on X, and each hyperplane H
is convex, each of the induced actions GH y N (H) is cocompact. Now by
convexity of N (H) and Proposition 2.2.6, B1 (H) consists of the join of ∆H
with a single vertex corresponding to H. Since X is finite-dimensional, we
43
may induct on dimension and assume that GH acts cocompactly on ∆H and
therefore on B1 (H). Since GHi acts cocompactly on B1 (Hi ) for 1 ≤ i ≤ k, and
(∪
)
∆X = G i B1 (Hi ) , the action of G on ∆X is cocompact.
Let Xr be the recubulation of X. By Lemma 2.4.1, G acts cocompactly
on Xr , and hence on ∆Xr by the previous argument. But X ⊂ Xr is a
G-invariant subcomplex, and the inclusion X ,→ Xr induces an equivariant
isomorphism ΓX ∼
= ∆Xr . Hence G acts cocompactly on ΓX.
Conversely, suppose that G acts properly on X and cocompactly on
ΓX. Let K be a finite subgraph of ΓX such that GK = ΓX. By possibly
adding finitely many vertices and edges to K, we may assume that K is a
full subgraph of ΓX and K 0 corresponds to an inseparable set of hyperplanes
containing a maximal clique. Indeed, each maximal clique in ΓX is finite since
X, being proper and cocompact, has finite degree.
By Proposition 2.3.4, there is a convex subcomplex F ⊂ X such that
the subgraph of ΓX induced by W(F ) is equal to K. Since the contact graph
K of F is finite, F is compact. Since K contains a maximal clique, F is the
unique convex subcomplex with contact graph K. Indeed, if F ′ were some
different subcomplex with the same contact graph, then any hyperplane W
separating F ′ from F would cross every element of W(F ), contradicting
maximality of the clique in K.
Let x ∈ X0 . Then x lies in the carrier of some hyperplane, so there
exists W ∈ W(F ) and g ∈ G such that gx ∈ N (W ), since W(F ) contains
a representative of each orbit of hyperplanes. Now GW acts properly on
N (W ) and cocompactly on ΓW . Hence, by induction on dimension, GW
acts cocompactly on N (W ). Thus there exists h ∈ GW such that x ∈
g −1 h−1 N (W ) ∩ F . Therefore, X = GF , whence G acts cocompactly on X.
Properness of the action is necessary in Proposition 2.4.2, as is shown by
the following example.
Example 2.4.3. Let X consist of a 0-cube v, together with a 0-cube xn for
each n ∈ Z, with a 1-cube joining each xn to v, so that X is an infinite star.
44
Let G ∼
= ⟨a⟩ ∼
= Z act by: av = v and axn = xn+1 . The action is cocompact,
since there is a single orbits of 1-cubes. Since Gv ∼
= Z, the action is not
proper. ΓX is the clique on the vertex set {Hn }n∈Z , where Hn is the midpoint
of the 1-cube joining v to xn .
Figure 2–5: Above is the compact quotient of X coming from the Z-action
in Example 2.4.3. Below is the part of the induced non-compact quotient of
contact graphs.
G acts on ΓX with a single orbit of vertices, since G acts transitively on
the 1-cubes of X, but there are infinitely many orbits of edges. Indeed, the set
{H0 ⌣
⊥ Hn }n∈Z is a single orbit of edges in Γ, but there are infinitely many
other such orbits, since the edge Hm ⌣
⊥ Hn can be assigned a “length” equal to
|m − n|, and Hm ⌣H
⊥ n = ak (Hs ⌣H
⊥ t ) only if |m − n| = |s − t|. See Figure 2–5.
The proof of Proposition 2.4.2 fails because ∆X is a discrete set of vertices
containing infinitely many G-orbits.
2.4.4
Essential actions
Let G act on X. Following [Sag95, CS11], we say that the hyperplane
H ∈ W is G-essential if there exists x ∈ X0 such that for all n ≥ 0, each
halfspace associated to H contains a point gx in the orbit Gx such that
dX (gx, N (H)) ≥ n. The action of G on X is essential if every hyperplane is
G-essential. If X is G-essential in this sense, then X is essential. In particular,
if X is G-essential then X is leafless. We therefore refer to an essential action
of G on the CAT(0) cube complex X as a leafless action, to emphasize
45
the fact that, if G acts essentially, then each halfspace associated to each
hyperplane properly contains a hyperplane.
The following very useful result of Caprace-Sageev allows us to assume,
given a cocompact cubulation, that the action of G is leafless, by passing if
necessary to a subcomplex:
Theorem 2.4.4 (Essential core theorem [CS11]). Let G act properly and
cocompactly on X. Then there exists a convex, G-invariant subcomplex Y ⊆ X
such that G acts leaflessly and cocompactly on Y.
The essential core theorem is also proven under other hypotheses
in [CS11]: the essential core Y is also nonempty provided G does not fix
a point on the visual boundary of X (even in the absence of cocompactness).
2.4.5
Unambiguous actions
Let G act on X. In Chapter 4, we will make an argument in which the
following situation occurs: there are two distinct hyperplanes, U and V ,
such that GU ∩ GV is infinite. We would like to conclude from this that the
collection of hyperplane stabilizers is not almost malnormal, but this would be
incorrect. Indeed, it could happen that U and V belong to distinct G-orbits,
but GU = GV . Hence GU ∩ GV is not of the form GU ∩ GhV , with h ̸∈ GV . The
following definition provides an additional hypothesis circumventing this issue.
Definition 2.4.5 (Unambiguous action). Let G act on the CAT(0) cube
complex X. Let S be the set of conjugacy classes of hyperplane-stabilizers
and let W be the set of orbits of hyperplanes. There is a surjection W → S
that sends the orbit of the hyperplane W to the conjugacy class of GW . If this
map is a bijection, then G acts unambiguously on X. Otherwise the action
is ambiguous, i.e. there exist hyperplanes U, V , in distinct orbits, such that
GU = G V .
Problem 2.4.6 (Removing ambiguity). Let G act properly and cocompactly
on X, with set W of hyperplanes and set S of conjugacy classes of hyperplane
46
stabilizers. Does there exist a cube complex Y such that G acts properly, cocompactly, and unambiguously on Y, in such a way that the set of hyperplane
stabilizers is again S?
If one is willing to forgo properness, then Y can be constructed using a
restriction quotient. Indeed, let O be a set of hyperplanes in X containing
a single hyperplane stabilized by each of the hyperplane-stabilizers in G.
Let Y be the restriction quotient obtained by restricting to the set GO of
hyperplanes. Then G acts unambiguously on Y, and it is not hard to show
(for example, by considering the induced quotient G\X → G\Y) that G acts
cocompactly on Y. However, the following example shows that this action
need not be proper.
Let X be the nonpositively-curved cube complex obtained from a single
2-cube by identifying all of its 0-cubes. Let F ∼
= F3 be the fundamental
group of X, and let X be the universal cover, shown in Figure 2–6. There
are two orbits of hyperplanes, and each is trivially stabilized as F acts by
deck transformations. Hence the action is ambiguous. If we form a restriction
quotient by eliminating one of these orbits, the resulting cube complex is
F -cocompact but, being locally infinite, is not F -proper.
Figure 2–6: At left is X, with two orbits of hyperplanes. Collapsing one orbit
yields a locally infinite cube complex, since there are pairs of 0-cubes arbitrarily far apart that are separated by blue hyperplanes only.
47
On the other hand, let F be a finitely generated nonabelian free group,
acting properly and cocompactly on a tree. In general, this action is ambiguous, since the hyperplane-stabilizers are all trivial and there is more than one
orbit of hyperplanes. However, in [Wisb], Wise showed that F acts properly
and cocompactly on a CAT(0) cube complex, with a single orbit of hyperplanes. In fact, for any finitely-generated infinite-index subgroup H ≤ F , such
an action can be constructed in which the hyperplane-stabilizers are the conjugates of H. In view of this fact, one approach to solving Problem 2.4.6 might
be to try to generalize Wise’s argument. For our purposes, ambiguity is only
an issue in Theorem 4.2.3, and hence for the moment we leave this situation
aside and merely hypothesize an unambiguous action where necessary.
The cube complexes provided by Sageev’s theorem (Theorem 2.4.7)
have hyperplanes stabilized by conjugates of a given codimension-1 subgroup
H. Now, if there is some H-invariant subset A ⊂ G that divides G into
exactly two complementary components, then there is a canonical way to
cubulate: each wall is essentially a partition into the two complementary
components of (some translate of ) A, and the action on the dual cube
complex is unambiguous since there is a single orbit of hyperplanes. On the
other hand, if A has many complementary components, these must be grouped
in some way when declaring the walls. One might imagine choosing two such
groupings, and thus two walls, each stabilized by H, whose corresponding
hyperplanes in the dual cube complex lie in different orbits.
2.4.6
Examples of cubulations
As alluded to earlier, the main source of actions on cube complexes is the
following theorem of Sageev:
Theorem 2.4.7 (Sageev [Sag95]). The finitely generated group G acts
essentially on a CAT(0) cube complex X if and only if there is a codimension1 subgroup H ≤ G. In such a case, H stabilizes a hyperplane. Moreover, if G
acts on a finite-dimensional CAT(0) cube complex, with no global fixed point,
then G has a codimension-1 subgroup.
48
Remark 2.4.8. We note in passing, but shall very rarely use, the fact that
the action of G on X is isometric with respect to the piecewise-Euclidean
CAT(0) metric dX [Gro87, Sag95].
Many well-known classes of groups are known to act on CAT(0) cube
complexes: finitely generated Coxeter groups act properly and with finitely
many orbits of hyperplanes on CAT(0) cube complexes [NR03]; Artin groups
of type FC act on finite-dimensional CAT(0) cube complexes [CD95b];
diagram groups, and in particular Thompson’s group V , act on CAT(0)
cube complexes – the latter action is proper and has two orbits of hyperplanes [Far03, Far05]; finitely presented B(4) − T (4) small-cancellation groups
act properly and cocompactly on cube complexes, and finitely-presented B(6)
groups act properly and with finitely many orbits of hyperplanes (though not
necessarily cocompactly) on cube complexes [Wis04]. As discussed below,
finitely generated right-angled Artin groups act properly and cocompactly on
CAT(0) cube complexes [CD95a].
The property of acting on a CAT(0) cube complex is generic, in the
sense that, in Gromov’s density model, random groups at density <
1
6
act
freely and cocompactly on CAT(0) cube complexes, and random groups at
density <
1
5
have codimension-1 subgroups [OW11]. By contrast, random
groups at density >
1
3
– and probably2 at density >
1
4
– have Kazhdan’s
property (T), since they satisfy Żuk’s beautiful spectral gap criterion [Żuk03].
Groups with property (T) cannot have codimension-1 subgroups, by a result
of Niblo-Roller [NR98], and it is interesting to ask what happens at density in
[ 15 , 14 ].
Theorem 2.4.7 generalizes the classical Bass-Serre theorem about actions
on trees. Indeed, if G splits as a graph of groups, then each vertex group and
edge group is a codimension-1 subgroup. On the other hand, G acts on a
2
Personal communication from Piotr Przytycki, April 2011.
49
1-dimensional CAT(0) cube complex – the Bass-Serre tree – in such a way
that the hyperplane-stabilizers are the (codimension-1) edge groups [Ser80].
From the point of view of random groups, actions on CAT(0) cube complexes
really do generalize splittings: random groups at sufficiently small density are
cubulated, but random groups at any positive density do not split [DGP10].
There is a close relationship between such splittings and the more general
“high-dimensional Bass-Serre theory” arising from Sageev’s theorem and
discussed in [Nib04a, Nib04b, NS, Sag97], among other places. A major
component of the relationship between splittings and cubulations comes
from recent work of Wise on groups with quasiconvex hierarchies. The basic
objects in Wise’s theory are special cube complexes, defined by HaglundWise in [HW08]. A nonpositively curved cube complex is special if it does
not contain certain pathological configurations of immersed hyperplanes;
equivalently, X is special if it admits a local-isometry to the Salvetti complex
of a right-angled Artin group. The class of groups containing finite-index
subgroups that are fundamental groups of special cube complexes is very
large, including Coxeter groups [HWa], one-relator groups with torsion [Wisa],
fundamental groups of Haken hyperbolic 3-manifolds [Wisa] and hyperbolic
limit groups [Wisa]. In [Wisa], Wise proves that, under various hyperbolicity
conditions, a group G is virtually the fundamental group of a special cube
complex if it admits a decomposition into trivial subgroups by a finite
sequence of splittings along quasi-isometrically embedded subgroups; this is a
quasiconvex hierarchy.
The characterization of special cube complexes in terms of right-angled
Artin groups is particularly attractive, and since we shall later discuss
the geometry of right-angled Artin groups, we now define the associated
nonpositively-curved cube complexes. This construction is due to CharneyDavis [CD95a], and we follow the exposition given in [Cha07].
50
A finitely generated right-angled Artin group is a group AΘ , where Θ is a
finite simplicial graph, presented by
⟨
⟩
AΘ = {av : v ∈ V(Θ)} | {[av , aw ] : (v, w) ∈ E(Θ)} ,
where V(Θ), E(Θ) are the vertex and edge sets of Θ. The class of right-angled
Artin groups interpolates between free groups (which arise when Θ is totally
disconnected) and free abelian groups (when Θ is a clique). Right-angled
Artin groups have numerous useful properties inherited by their subgroups,
and hence by all special groups, notably Z-linearity [DJ00, HW99] and the
RFRS property defined by Agol in [Ago08], which characterizes fundamental
groups of virtually fibered 3-manifolds.
The presentation complex associated to the above presentation for AΘ
has a single 0-cell x0 , with an oriented loop av corresponding to each vertex
v ∈ V(Θ). For each relator, corresponding to an edge (v, w) ∈ E(Θ), we attach
−1
a 2-cube according to the word av aw a−1
v aw . This yields the presentation
complex XΘ2 , which is a (possibly positively curved) compact 2-dimensional
cube complex. The simplicial complex lk(x0 ) has a 0-cell v − for the incoming
edge av , and a 0-cell v + for the outgoing edge av , for each v ∈ V(Θ). The
0-simplices v ± , w± are joined by a 1-simplex exactly when [av , aw ] commute.
For each n-clique in lk(v)1 , with n ≥ 3, we see that the n corresponding
generators pairwise-commute, and we add an n-cube to XΘ2 wherever its 2skeleton appears, so that the corresponding clique in the link spans a simplex.
In this way we obtain a compact nonpositively-curved cube complex XΘ with
π 1 XΘ ∼
= AΘ , so that AΘ acts freely and cocompactly on the CAT(0) cube
eΘ that is the universal cover of XΘ . XΘ is the Salvetti complex of
complex X
AΘ . The generating graph Θ of AΘ is closely related to the crossing graph of
eΘ .
X
51
CHAPTER 3
Weak hyperbolicity and quasi-arboreal groups
Section 3.1 is devoted to establishing Theorem 3.1.1, which says that
there exist constants M, C < ∞, independent of X, and a tree T, such that
there is an (M, C)-quasi-isometry ΓX → T . The author first proved this
in [Hag11], where the bound M ≤ 4, C = 0 is obtained using the method
of disc diagrams with fixed carriers; a slight simplification of the original
argument still yields M ≤ 5, as is also noted in [CH11]. These disc diagram
proofs construct T explicitly, and reflect the characterization of graphs quasiisometric to trees given by Krön and Möller in [KM08]. However, in [Man05],
Manning introduced another criterion – the bottleneck property – guaranteeing
that a given geodesic space is quasi-isometric to a tree:
Proposition 3.0.9 (Bottleneck criterion [Man05]). Let (Y, d) be a geodesic
metric space. Then there exists a simplicial tree T , constants M, C, and an
(M, C)-quasi-isometry Y → T if and only if there exists δ > 0 such that, for
all x, y ∈ Y, there is a midpoint m = m(x, y) such that
1
d(m, x) = d(m, y) = d(x, y),
2
with the property that any path γ : [a, b] → Y joining x to y satisfies
d(γ(t), m) < δ for some t ∈ [a, b].
At the expense of larger M, C Theorem 3.1.1 can be established by a
significantly simpler argument, also given in [Hag11], using the bottleneck
property, and this is the proof given in Section 3.1. The original proof is quite
technical, so we content ourselves with a sketch.
Section 3.2 concerns weak hyperbolicity of cubulated groups. From
Theorem 3.1.1, it follows that groups acting on CAT(0) cube complexes act
on quasi-trees, and we give conditions under which this implies that the group
52
G acting on X is weakly hyperbolic relative to the collection of hyperplane
stabilizers. The weak hyperbolicity is actually of a specialized sort – the
G-graph required by Bowditch’s definition is not only Gromov-hyperbolic, but
actually quasi-isometric to a tree. The group G is thus defined to be quasiarboreal relative to the collection of hyperplane-stabilizers, and we conclude
the chapter with a few examples of quasi-arboreal groups in which the defining
G-quasi-tree cannot be the contact graph of a proper G-cube complex.
3.1
Contact graphs are quasi-trees
Theorem 3.1.1. There exist constants M, C < ∞ such that, for all CAT(0)
cube complexes X, the contact graph ΓX is (M, C)-quasi-isometric to a tree.
Proof. Let V0 , Vn be hyperplanes of X. Then either V0 ⌣
⊥ Vn , or V0 = Vn ,
or there exists a set V1 , . . . , Vn−1 containing exactly those hyperplanes that
separate V0 from Vn , labeled so that, if Vi , Vk are separated by Vj , then
i < j < k or k < j < i. By definition, we have V0 ⌣
⊥ V1 ⌣
⊥ ... ⌣
⊥ Vn−1 ⌣
⊥ Vn . In
the first case, let m = m(V0 , Vn ) be the midpoint of the edge V0 ⌣V
⊥ 1 of ΓX. In
the second case, let m = m(V0 , Vn ) = V0 = Vn . In the third case, if n = 2k for
some k ≥ 1, then let m = m(V0 , Vn ) = Vk . Otherwise, m = 2k + 1 for some k,
and m = m(V0 , Vn ) is the midpoint of the edge Vk ⌣V
⊥ k+1 .
Let σ : [a, b] → ΓX be a path with σ(a) = V0 , σ(b) = Vn . Then, for
each hyperplane Vi separating V0 from Vn , the path σ must pass through Vi
or through a vertex corresponding to a hyperplane that crosses Vi . Hence,
for any δ > 23 , the path σ comes within δ of m. See Figure 3–1. It follows
from Proposition 3.0.9 that there is a tree T and an (M, C)-quasi-isometry
ΓX → T, for some M ≥ 1, C ≥ 0. From Manning’s proof of the bottleneck
criterion, we compute the quasi-isometry constants; these depend only on the
fact that δ >
3
2
and are in particular independent of X.
The conclusion of the proof of the bottleneck criterion in [Man05] is that
there exists a map β : T → ΓX such that for all vertices s, t ∈ T , we have
8δdT (s, t) − 16δ ≤ dΓX (β(s), β(t)) ≤ 26δdT (s, t),
53
Figure 3–1: The arrowless hyperplanes are those separating V0 from Vn . The
arrowed hyperplanes form a path in the contact graph from V0 to Vn . Every
separating hyperplane either belongs to this path or crosses at least one of the
hyperplanes on this path.
and β is 20δ quasi-surjective. It follows that ΓX is (M, C, K)-quasi-isometric
to a tree for any M > 39, C > 24, K > 30.
Example 3.1.2. If X ∼
= X1 × X2 is a cubical product such that Xi
contains at least one hyperplane for i ∈ {1, 2}, then ΓX is a nontrivial
join, by Proposition 2.3.7. Hence diam(ΓX) ≤ 2, from which it follows that
ΓX is quasi-isometric to a tree with one vertex. In particular, if X is the
Salvetti complex of a right-angled Artin group AΘ , where Θ decomposes as a
nontrivial join, then ΓX has finite diameter.
If X is a tree, then we saw that ΓX is the graph dual to X. Let T be
the first barycentric subdivision of X, with the standard path-metric. Then
there is a map q : ΓX → T that sends each vertex H to the vertex of T
corresponding to the midpoint of the 1-cube of X dual to H. A computation
shows that dT (q(V ), q(H)) = 2dΓX (V, H) for all V, H, and hence q is a
quasi-isometry.
Remark 3.1.3 (Grading hyperplanes). Let Γ be a graph, and fix a base
vertex V0 . With respect to V0 , the grade g(V ) of the vertex V is dΓ (V, V0 ).
The full sphere Sn is the full subgraph of Γ generated by the grade-n vertices,
54
for n ≥ 0, and the full ball Bn is the full subgraph generated by the vertices of
grade at most n.
Let V, V ′ ∈ Sn be vertices. Declare V ∼ V ′ if and only if V, V ′ are joined
by a path P in Γ such that every vertex of P has grade at least n. This is
an equivalence relation on the grade-n hyperplane; the full-subgraph Cn (V )
generated by the equivalence class of V is a grade-n cluster.1
The cluster tree T associated to Γ, V0 is the tree whose vertex-set is the
set of all clusters of all grades. The clusters Cn (V ), Cn−1 (W ) are adjacent in
T exactly when there exists V ′ ∼ V and W ′ ∼ W such that V ′ and W ′
are adjacent in Γ. If Cn (V ) is a grade-n cluster, with n ≥ 1, then Cn (V ) is
adjacent to a unique grade-(n − 1) cluster, by the definition of the relation ∼.
It follows immediately that T is in fact a tree.
Moreover, there is a natural surjection Γ → T that sends each vertex
V to the unique cluster containing it, each edge (V, W ) with V ∼ W to the
vertex Cn (V ) = Cn (W ), and each edge (V, W ) with V ∈ Cn (V ), W ∈ Cn+1 (W )
to the edge of T joining the corresponding clusters. A heuristic picture
of a graded graph Γ, the cluster tree T , and the map Γ → T is shown in
Figure 3–2.
Suppose there exists M ≥ 1 such that diamΓ (C) ≤ M for all clusters C.
Then a simple computation shows that Γ → C is an (M, 0) quasi-isometric
embedding, and thus a quasi-isometry, being surjective. Krön and Möller
independently proved a very similar result characterizing graphs quasiisometric to trees [KM08].
In [Hag11], this argument is applied to the contact graph ΓX of X: we
grade hyperplanes from a fixed base hyperplane according to their distance
in the contact graph, and verify that no cluster has diameter greater than
1
In [Hag11], it is somewhat randomly called a root; when we later used
the same concept in [CH11], V. Chepoi suggested the superior term “cluster”.
Words matter.
55
Figure 3–2: At left is the graded graph Γ, partitioned into grades (indicated
by the translucent colored annuli), which are further partitioned into clusters.
At right is the corresponding cluster tree.
4; establishing this bound is the main part of the proof using disc diagrams.
More specifically, given hyperplanes V1n , V2n in the same grade-n cluster,
we choose geodesics in ΓX joining V1n , V2n to the base hyperplane V0 , and
a shortest path joining V1n to V2n that does not contain any hyperplane of
grade less than n. Each of these paths is realized in X in the following way:
if U1 ⌣
⊥ U2 ⌣
⊥ ... ⌣
⊥ Uk is a path in ΓX, then there is a concatenation
P1 P2 . . . Pk−1 of geodesic segments Pi → N (Ui ). Hence the triangle in
ΓX determined by the given paths is represented by a closed path in ΓX,
which bounds a disc diagram D → X. We choose the paths in ΓX and
their representative paths in X to be as short as possible, and, moreover, to
minimize the area of D among all possible disc diagrams formed in this way.
A heuristic picture of the resulting diagram appears in Figure 3–3. One then
examines dual curves in D, eliminating possible endpoints using the various
minimality assumptions together with hexagon moves. Arguing by induction
on grade, one concludes that dΓX (V1n , V2n ) ≤ M, where M = 4 or 5, depending
on the particular argument used. Grading the vertices in ΓX and in ∆X also
plays an important role in Chapter 6, although in that context we do not use
disc diagrams or cluster trees.
56
Figure 3–3: The diagram D used in the cluster-tree proof of Theorem 3.1.1.
As is often convenient when considering osculations, we draw hyperplanecarriers rather than hyperplanes.
3.2
Applications to cubulated groups
The notion of the contact graph arose in an attempt to answer the ques-
tion of for which cocompact cubulations of a group G the pair (G, {GH }) is
relatively hyperbolic, where {GH } is the collection of hyperplane-stabilizers.
In this section, we discuss the implications of Theorem 3.1.1 for weak hyperbolicity in the sense of Bowditch-Farb, as well as introducing the stronger notion
of quasi-arboreality.
3.2.1
Weak hyperbolicity
Let G be a finitely generated group, let {Pi }i∈I be a finite collection of
finitely-generated subgroups, and let Λ be a locally finite Cayley graph of
G. The coned-off Cayley graph of the pair (G, {Hi }) was defined by Farb
in [Far98], as follows: for each left coset gHi , add a vertex v(gHi ) to Λ, and
join v(gHi ) by an edge to each vertex of Λ corresponding to an element of
b Farb declared G to
gHi . The resulting graph is the coned-off Cayley graph Λ.
b is δ-hyperbolic for some δ < ∞. This nobe hyperbolic relative to {Hi }i∈I if Λ
tion does not coincide with the notion of relative hyperbolicity introduced by
Gromov in [Gro87] and discussed by Bowditch [Bow97] and others. Instead,
57
Farb’s criterion is equivalent to the definition of weak hyperbolicity of G relative to {Hi }i∈I given by Bowditch; to pass from weak hyperbolicity to relative
hyperbolicity, Farb imposes an additional constraint called bounded coset penetration. Since Bowditch’s characterization of relative hyperbolicity provides
an attractive way of circumventing this somewhat technical definition, we shall
not discuss bounded coset penetration.
One can substitute any proper, cocompact G-space for the Cayley graph
in Farb’s construction. Hence consider the situation in which G acts properly
and cocompactly on the CAT(0) cube complex X, and let W be a finite set of
hyperplanes containing exactly one hyperplane from each orbit. Let W = GW
be the set of all hyperplanes. In analogy to Farb’s construction, we form the
coned of CAT(0) cube complex as follows. For each hyperplane H ∈ W, the
corresponding coned-off hyperplane is the quotient:
K(H) =
[0, 1] × N (H)
.
{1} × N (H)
We then form the coned-off cube complex K(X) by identifying N (H) with
N (H) × {0} ⊂ K(H) according to the map N (H) ∋ x 7→ (x, 0) ∈ N (H) × {0},
for each H ∈ W. The 1-skeleton K(X)1 consists of X1 , together with a conevertex v(H) for each H ∈ W, with a cone-edge joining v(H) to each 0-cube of
N (H). Note that K(X) is a cubical presentation in the sense of [Wisa], related
to, but different from, that in which G\X is the nonpositively curved cube
complex of “generators”, and the “relators” are the immersed hyperplanes.
G acts on K(X): the original cube complex X is a G-invariant subcomplex, cone-vertices are sent to cone-vertices, and cone-edges to coneedges. Moreover, the stabilizer of the cone-vertex v(H) is precisely GH . In
other words, K(X) is very much like a coned-off Cayley graph for the pair
(G, {GW }).
By construction, the vertices v(H), v(H ′ ) are non-adjacent in K(X)1 for
H, H ′ ∈ W, since every cone-edge contains exactly one cone-vertex. However,
if there exists x ∈ N (H)0 ∩ N (H ′ )0 , then there is a concatenation of two
58
cone-edges that passes through x and joins v(H) to v(H ′ ). Consider the graph
whose vertex set is {v(H) : H ∈ W}, with v(H), v(H ′ ) joined by an edge if
and only if dK(X)1 (v(H), v(H ′ )) = 2. This graph is exactly ΓX! Indeed, the
assignment v(H) 7→ H defines a bijection on vertices. On the other hand, any
path of length 2 joining v(H) to v(H ′ ) contains at least two cone-edges and
thus passes through some x ∈ N (H)0 ∩ N (H ′ )0 , whence H ⌣
⊥ H ′ . Conversely,
if H ⌣
⊥ H ′ , then such a path exists. It follows that there is a quasi-isometric
embedding ΓX → K(X)1 that is 1-quasi-surjective, since each 0-cube of X lies
in at least one hyperplane-carrier. By Theorem 3.1.1, the “coned-off Cayley
graph” K(X)1 is therefore a quasi-tree.
This construction can be done more generally, by choosing some space
on which G acts geometrically and coning off geometric representative of the
peripheral subgroups. This motivates the following definition.
Definition 3.2.1 (Weak hyperbolicity [Bow97]). Let G be a finitely generated
group and let {Hi }i∈I be a finite collection of subgroups. Let G act by
isometries on a graph Γ, satisfying the following conditions:
1. Γ is hyperbolic.
2. G acts with finitely many orbits of edges.
3. For each vertex V of Γ, the stabilizer of V contains a finite-index
subgroup that is conjugate to some Hi .
4. Each Hi stabilizes a vertex of Γ.
Then G is weakly hyperbolic relative to the collection {Hi }i∈I of peripheral
subgroups.
A stronger property is:
Definition 3.2.2 (Quasi-arboreality). Let G, {Hi }i∈I be as in Definition 3.2.1,
and let Γ be a G-graph satisfying conditions (2), (3), (4) of Definition 3.2.1. If
Γ is quasi-isometric to a tree, then G is quasi-arboreal relative to {Hi }i∈I .
Since trees are 0-hyperbolic, the pair (G, {Hi }i∈I ) is weakly hyperbolic
if it is quasi-arboreal. The main result of this section says that cocompactly
cubulated groups are quasi-arboreal:
59
Corollary 3.2.3. Any group acting on a CAT(0) cube complex acts by
isometries on a quasi-tree. Furthermore, let G ∼
= π1 X be the fundamental
group of a compact nonpositively-curved cube complex, and let W be the set
of immersed hyperplanes in X. For each W ∈ W, let GW ≤ G be the image
of π1 W under the monomorphism induced by the local isometry N (W ) → X.
Then G is quasi-arboreal relative to {GW : W ∈ W}.
Proof. The first assertion is immediate from Theorem 3.1.1. To prove the
second assertion, let X be the universal cover of X, with contact graph
ΓX. Then G acts by isometries on ΓX in such a way that the set of vertexstabilizers of ΓX coincides with the set of stabilizers of hyperplanes in X,
i.e. with the set of conjugates of the subgroups GW . Since G acts properly
on X, the action on ΓX has finitely many orbits of edges since X is compact,
by Proposition 2.4.2. Finally, ΓX is a quasi-tree. Thus each requirement of
Definition 3.2.2 is satisfied.
The question of relative hyperbolicity, in the stronger sense of Gromov
and Bowditch, of the pair (G, {GW }) is taken up in Chapter 4, where we
study fineness of the contact graph.
Corollary 3.2.3 seems analogous to results of Charney-Crisp on weak
hyperbolicity of Artin groups: in [CC07], they give conditions under which
the Artin group A is weakly hyperbolic relative to its finite type standard
parabolic subgroups. If A is a right-angled Artin group, then the hyperplanestabilizers are finite type standard parabolic subgroups, although the converse
is not true. Charney and Crisp work in a manner analogous to our proof of
Corollary 3.2.3, using the Deligne complex in roughly the way that we have
used the contact graph.
3.2.2
Examples of quasi-arboreal groups
By Corollary 3.2.3, each of the groups listed in Section 2.4 acts by
isometries on a quasi-tree, and each of the groups for which the action
is proper and cocompact – finitely generated right-angled Artin groups,
60
B(4) − T (4) small-cancellation groups, etc. – is quasi-arboreal relative to the
collection of hyperplane-stabilizers.
On the other hand, the following examples show that a group can be
quasi-arboreal relative to a sensible collection of subgroups while failing to be
cubulated:
Example 3.2.4 (Baumslag-Solitar groups). Let F be a finitely-generated
free group, and let G ∼
= N o F for some finitely-generated subgroup N. Then
there is a graph Γ, whose vertices correspond to left cosets of N , with gN and
hN joined by an edge if they differ by multiplication by a (free) generator of
F ∼
= G/N. Then Γ is a Cayley graph for F with respect to a free generating
set, and Γ is therefore a tree. On the other hand, G acts with finitely many
orbits of edges on Γ in such a way that Stab(v) = N for each vertex v ∈ Γ.
Hence G is quasi-arboreal relative to N.
Let G ∼
= ⟨a, b | bam b−1 a−n ⟩, where m ̸= ±n. This G is not cubulated, by a
result of Haglund [Hag07]. However, the map G → Z given by a 7→ 0, b 7→ 1 is
an epimorphism to a free group, with cyclic kernel, so that G is quasi-arboreal
relative to a cyclic subgroup.
Example 3.2.5 (Special linear groups). Consider the Steinberg presentation
for SLn (Z), with n ≥ 3, where the generator aij represents the n × n matrix
with diagonal entries equal to 1, the ij-entry equal to 1, and 0 elsewhere:
⟨
⟩
−1 4
SLn (Z) ∼
= aij , 1 ≤ i ̸= j ≤ n | [aij , akl ], i ̸= k, j ̸= l; [aij , ajk ]a−1
ik , i ̸= k; (a12 a21 a12 ) .
Let Aij = ⟨aij ⟩ and denote by Γ the coned-off Cayley graph of the pair
(SLn (Z), {Aij }). A theorem of Carter and Keller implies that SLn (Z) is
boundedly generated with respect to {Aij } [CK83]. The graph Γ is therefore
bounded, and hence SLn (Z) is quasi-arboreal relative to {Aij }. On the other
hand, SLn (Z) has Property (T) [dlHV89] and thus contains no codimension-1
subgroups [NR98].
Example 3.2.6 (Osin’s examples). In [Osi05], Osin produced a class of
groups G with the following properties: G is weakly hyperbolic relative to a
61
finite collection P of infinite cyclic subgroups; the Bowditch graph for the pair
(G, P) has finite diameter; G contains every recursively presentable group.
The first two properties guarantee that G is quasi-arboreal relative to P. The
third property shows that G contains SL3 (Z), and therefore does not act
properly on a CAT(0) cube complex, since SL3 (Z) is infinite and has property
(T).
3.2.3
Problems on weak hyperbolicity and quasi-arboreality
Problem 3.2.7 (Weak hyperbolicity). Let G be an arbitrary group acting
nontrivially on a CAT(0) cube complex X. Is there a finite collection S of
subgroups of G such that G is weakly hyperbolic relative to S? Corollary 3.2.3
says that the answer is yes if the action is proper and cocompact; S is the set
of hyperplane-stabilizers (up to conjugacy). Moreover, if G ∼
= H∗At =B , then
G is weakly hyperbolic relative to H, and if G ∼
= H ∗A=B K, then G is weakly
hyperbolic relative to the pair {H, K}, for arbitrary groups H, K, by a result
of Osin [Osi04].
However, one cannot in general choose S to be the set of hyperplanestabilizers. The most obvious failure is an unambiguous action with infinitely
many orbits of hyperplanes; the action on the contact graph cannot be
cocompact in this situation. More generally, G could act with finitely many
orbits of hyperplanes, but fail to act cocompactly on ΓX. It may be that
such a weakly hyperbolic structure does not in general exist, or that one must
choose less obvious peripheral subgroups.
Problem 3.2.8 (Quasi-arboreality and splittings). Let G be a wordhyperbolic group acting properly and cocompactly on the CAT(0) cube
complex X. We may hypothesize a leafless action, by Theorem 2.4.4, and
hence, by Theorem 6.2.3 below and Proposition 2.4.2, ΓX is unbounded and
G-cocompact. Let V be a hyperplane and let B be the 1-ball in ΓX centered
at V . Does there exist a finite-index subgroup G′ ≤ G such that, for all
g ∈ G′ , either gB = B or gB ∩ B = ∅? A positive answer would imply
that the cube complex dual to the wallspace (X0 , G′ {V }) is a tree, and hence
62
that G virtually splits over GV . Note that this is somewhat plausible, since B
separates ΓX, by definition, and by cocompactness and unboundedness of ΓX,
there exists an “evenly distributed” family of pairwise disjoint translates of B.
Now, let G act on X, and let H be a hyperplane such that GH acts on
H with a global fixed point. Does G split over a subgroup commensurable
with GH ? This is the Kropholler-Roller conjecture [NS, KR], and it seems that
there may be an approach using the “block tree decomposition” of ΓX.
More speculatively, let G be quasi-arboreal relative to a finite set S of
subgroups. Suppose that the implied G-quasi-tree is unbounded. Under what
conditions does G (virtually) split over a subgroup (commensurable with)
S ∈ S?
63
CHAPTER 4
Bicliques in the contact graph
The complete bipartite subgraphs of ∆X and ΓX encode useful information about X. In Section 4.1, we give a condition on such subgraphs that
characterizes hyperbolic cube complexes. In Section 4.2, we use similar techniques to characterize fineness of the contact graph and thus hyperbolicity of
cocompactly cubulated groups relative to the hyperplane stabilizers.
4.1
Thin bicliques and the cubical flat plane theorem
The goal of this section is to prove Theorem 4.1.3, which says that a
finite-degree cube complex is hyperbolic if and only if its crossing graph
has thin bicliques. From this we deduce Theorem 4.1.3, which is a cubical
analogue of the flat plane theorem (see [BH99]), stated in terms of the
crossing graph. Since our proof relies on establishing super-linearity of the
isoperimetric function in the absence of the thin bicliques condition, we are
actually appealing, indirectly, to the flat plane theorem for CAT(0) spaces.
This section coincides with the corresponding discussion in [Hag11], with some
small changes.
Definition 4.1.1 (Thin bicliques). The graph Γ has thin bicliques if there
exists n ∈ N such that any complete bipartite subgraph Kp,q ⊆ Γ satisfies
p < n or q < n.
Definition 4.1.2 (Grid). If I, J are subdivided intervals, then I × J is an
|I| × |J| grid, and X contains an |I| × |J| grid if there is a cubical isometric
embedding I × J → X.
Theorem 4.1.3. Let G be a group acting properly and cocompactly on the
CAT(0) cube complex X. Then G is word-hyperbolic if and only if ∆X has
thin bicliques. If G is not word-hyperbolic, then ∆X contains the complete
bipartite graph K∞,∞ .
64
Proof. G is quasi-isometric to X1 , and thus X1 is hyperbolic if and only if
G is word-hyperbolic. The first assertion follows from Theorem 4.1.6 below.
If G is not word-hyperbolic, then ∆X contains arbitrarily large complete
bipartite subgraphs. Let Λ be a graph whose vertices correspond to complete
j
i
i
bipartite subgraphs Kn,n
of ∆X, with Kn,n
adjacent to Kn+1,n+1
if and only if
j
i
Kn,n
⊂ Kn+1,n+1
. The graph Λ is infinite since ∆X contains arbitrarily large
complete bipartite graphs. Moreover, by cocompactness, there are finitely
many G-orbits of embedding of Kn,n in ∆X, for each n, so that Λ is locally
finite and has an infinite component. By Lemma 2.1.1, there is an increasing
family {Kn,n }n of complete bipartite subgraphs of Λ, and thus a subgraph
K∞,∞ ⊂ ∆X.
The following lemma collects basic facts about hyperbolicity of cube
complexes and the thin bicliques property of crossing graphs.
Lemma 4.1.4. For a CAT(0) cube complex X with crossing graph ∆, we
have:
1. If X is finite-dimensional, then X, with the usual CAT(0) metric, is
hyperbolic if and only if X1 is a hyperbolic graph.
2. If X is infinite-dimensional, then it is not hyperbolic, and neither is X1 .
3. If ∆X has thin bicliques, then X is finite-dimensional.
Proof. (1) follows from the fact that a finite-dimensional CAT(0) cube
complex is quasi-isometric to its 1-skeleton.
To prove (2), note that for any d ≥ 0, the existence of a d-cube guarantees the presence of a geodesic triangle, whose corners are 0-cubes, that is not
d-thin. Hence X is not d-thin for any d if X contains arbitrarily large cubes.
If ∆X has thin bicliques, then there is an upper bound on the cardinality
of cliques in ∆X, since the existence of a complete subgraph on 2d vertices
implies the existence of a complete (d, d) bipartite subgraph. The dimension of
X is the maximal cardinality of cliques in ∆X, and (3) follows.
65
When using disc diagrams, it is sometimes easier to think of a δhyperbolic space as one whose isoperimetric inequality is linear than it is
to verify the thin triangle condition. Hence we shall sometimes rely on the
following version of Gromov’s characterization of hyperbolic metric spaces as
those having linear isoperimetric inequality. This result is stated in cubical
terms as follows.
Lemma 4.1.5 (Linear isoperimetric inequality [Gro87]). Let X be a CAT(0)
cube complex that is δ-hyperbolic with respect to its CAT(0) metric. There
exists λ ≥ 0 such that for each closed combinatorial path σ → X, there exists a
disc diagram D → X with ∂p D = σ such that the area of D is at most λ|σ|.
Actually, only the fact that the isoperimetric function of a hyperbolic
metric space is subquadratic is invoked in our applications.
Theorem 4.1.6. The finite-degree CAT(0) cube complex X is hyperbolic if
and only if ∆X has thin bicliques.
Note that Theorem 4.1.6 implies that X is hyperbolic when the contactgraph has thin bicliques. The proof of Theorem 4.1.6 is assembled as follows
from the lemmas below.
Proof of Theorem 4.1.6. By Lemma 4.1.9, X is hyperbolic when ∆X has
thin bicliques. Conversely, if ∆X does not have thin bicliques, then by
Lemma 4.1.8, X does not have a linear isoperimetric function and thus, by
Lemma 4.1.5, X is not δ-hyperbolic for any δ.
Remark 4.1.7 (Planar grids from 4-cycles in ∆X). Let H0 ⊥V0 ⊥H1 ⊥V1 ⊥H0
be an embedded 4-cycle in ∆X. For i ∈ {0, 1}, choose concatenable geodesic
paths Pi → N (Vi ), Qi → N (Hi ) such that A = P0 Q0 P1 Q1 is a closed path
in X. Let D → X be a disc diagram with boundary path A. Suppose that A
and D are chosen so that D has minimal area among all such diagrams for the
given 4-cycle in ∆X. See Figure 4–1.
If K is a dual curve in D, then K travels from P0 to P1 or from Q0 to
Q1 . Indeed, K cannot travel from, say, P1 to P1 , since P1 is a geodesic. Now
66
Figure 4–1: The diagram arising from a 4-cycle in ∆X.
K cannot travel from Pi to Qj , for then we could modify A, without affecting
the 4-cycle in ∆X, to produce a lower area diagram from the same 4-cycle
in ∆X (see the description of diagrams with fixed carriers in [Hag11], or the
discussion of splaying and rectangles in [Wisa]). Similarly, no two dual curves
emanating from Pi (or Qi ) can cross. Denote by H the set of dual curves
traveling from P0 to P1 and by V the set of dual curves traveling from Q0
to Q1 . Each element of H crosses each element of V, and there are no other
intersections of dual curves in D. Hence D is a grid isomorphic to P0 × Q0 . In
particular, D is a CAT(0) cube complex whose set of hyperplanes is H ⊔ V.
Let H, H ′ ∈ H and V ∈ V be dual curves in D. Since H crosses V , the
dual curves H and V map to distinct hyperplanes of X, since hyperplanes
in X do not self-cross. Since H and H ′ are both dual to 1-cubes of P0 ,
and P0 → X is a geodesic, the dual curves H, H ′ must map to distinct
hyperplanes. Hence the map D → X is injective on hyperplanes. Since
D → X is a cubical map of CAT(0) cube complexes that is injective on
hyperplanes, it is an isometric embedding.
Suppose r ≤ min(|P0 |, |Q0 |). Then D contains an r × r grid E with
boundary path P . Note that the map D → X restricts to an isometric
67
embedding E → X, and in particular P embeds in X. Note that |P | = 4r,
while |E| = r2 .
If there exists a disc diagram F → X with ∂p F = P and Area(F ) < r2 ,
then we could excise the interior of E from D and attach F along P to obtain
a lower-area diagram D′ with boundary path A, contradicting the fact that D
has minimal area among diagrams with boundary path A. Hence every disc
diagram bounded by P has area at least r2 .
Lemma 4.1.8. Let X have degree D < ∞. If ∆X does not have thin
bicliques, then X is not δ-hyperbolic for any δ < ∞.
Proof. If the degree D of X is 0, then X is a 0-cube. If D = 1, then X is a
1-cube. If D = 2, then either X is a single 2-cube or X is an interval. In each
of these cases, ∆X has thin bicliques and X is hyperbolic.
If D = 3, then by Remark 4.1.7, ∆X cannot contain an embedded 4-cycle
and thus has thin bicliques. On the other hand, either X is a single 3-cube,
or X embeds in T × [− 12 , 12 ] for some tree T . Hence X is hyperbolic. Thus we
assume that D > 3.
For 2 ≤ R < ∞, let H, V be disjoint sets of hyperplanes, with
min(|H|, |V|) ≥ R, such that K(V, H) ⊆ ∆X, i.e. for all V ∈ V, H ∈ H, we
have V ⊥H. Let V0 , V1 be distinct hyperplanes in V and let H0 , H1 be distinct
hyperplanes in H. Then H0 ⊥V0 ⊥H1 ⊥V1 ⊥H0 is an embedded 4-cycle in ∆X.
Without loss of generality, V and H are inseparable. Indeed, if W is a
hyperplane separating H, H ′ ∈ H, then W crosses each V ∈ V, since V ⊥H
and V ⊥H ′ . Hence we can include W in H without affecting the fact that V
and H generate a biclique in ∆X.
Choose V0 ∈ V. For each H ∈ H, let H̄ = H ∩ V0 be the hyperplane of
V0 corresponding to H. Let H̄1 , H̄2 , . . . H̄R be an inseparable collection of R
distinct hyperplanes of V0 , numbered so that, for all i, j, k, if i < k and H̄j
separates H̄i from H̄j , then i < j < k.
By Proposition 2.3.4, there exists an isometrically embedded tree T ⊆
N (V0 ) ⊆ X such that the set of hyperplanes crossing T is precisely {Hi }R
i=1 .
68
Claim: There exists a geodesic P0 in T of length at least S = S(R),
where S(R) → ∞ as R → ∞.
Proof of Claim: Recall that D > 3. Each 0-cube t ∈ T has degree at
most D − 1 in T . Indeed, t has at most D incident 1-cubes in X, one of which
is dual to the hyperplane V0 that does not cross T . Let S be the length of a
longest geodesic segment P0 in T . By choosing a root of T , we construct a
rooted regular (D − 2)-ary tree T ′ , containing the rooted tree T , such that the
depth a of T ′ satisfies 2a ≤ S. T ′ has at least as many edges as T , so that
R≤
(D − 2)a+1 − 1
− 1,
D−3
from which it follows that
S ≥ 2 logD−2 [(D − 3)(R + 1) + 1] − 2,
which is increasing and unbounded as R → ∞.
Choose TH , TV to be isometrically embedded trees such that TH is crossed
exactly by a set of at least R elements of H and TV by at least R elements
of V. Since each H dual to a 1-cube of TH crosses each V dual to a 1-cube
of TV , we may apply Proposition 2.3.8 to produce an isometric embedding
TH × TV → X. By the Claim, there exist geodesic segments P0 ⊆ TH and
Q0 ⊆ TV , each of length at least S. Hence there is an isometric embedding
P → X, where P is an S × S grid in TH × TV . Let A → X be the closed
(embedded) path in X obtained by restricting the map P → X to the
boundary of P . Note that Area(P ) = S 2 and |A| = 4S. By Remark 4.1.7, any
disc diagram in X with boundary path A has area at least S 2 . Since S is an
increasing function of R, and R can be chosen arbitrarily large since ∆X does
not have thin bicliques, X does not satisfy the conclusion of Lemma 4.1.5 and
therefore is not hyperbolic.
Lemma 4.1.9. If ∆X has thin bicliques, then X is hyperbolic.
69
Proof. By Lemma 4.1.4, it suffices to show that X1 is δ-hyperbolic for some δ.
Suppose to the contrary that for any n ∈ N, there exists a combinatorial
geodesic triangle χn ηn νn → X1 that is not n-thin. Let Dn → X be a
disc diagram of minimal area with boundary path χn ηn νn . By assumption,
there exists a point x ∈ χn such that dX (x, ηn ∪ νn ) > n. Let V be the
set of hyperplanes separating x from νn and let H be the set of hyperplanes
separating x from ηn . Let V be the set of dual curves in Dn that separate x
from νn in Dn and let H be the set of dual curves in Dn separating x from ηn .
The diagram Dn is shown in Figure 4–2.
Figure 4–2: The diagram Dn and a some vertical and horizontal separating
dual curves.
We first show that the map Dn → X induces bijections V → V and
H → H and deduce that |V|, |H| ≥ n. A disc diagram argument then shows
that each element of V crosses each element of H and thus that K(V, H) is a
complete bipartite subgraph of ∆X with |V|, |H| ≥ n. Hence the failure of X
to be hyperbolic implies that ∆X does not have thin bicliques.
The correspondences between V, H and V, H: Dual curves in Dn
map to distinct hyperplanes. Indeed, since each side of the triangle ∂p Dn is a
geodesic segment, no dual curve has both endpoints on the same side, because
a geodesic contains at most a single 1-cube dual to each hyperplane. Hence, if
C, C ′ are distinct dual curves in Dn , then one of the sides χn , ηn , νn contains
two of the four endpoints of C ∪ C ′ . Thus C and C ′ cannot map to the same
hyperplane, for otherwise that side would cross a single hyperplane in two
70
distinct 1-cubes, contradicting the fact that it is a geodesic. Hence the maps
V, H → V, H that associate dual curves in Dn to hyperplanes according to the
map Dn → X are injective.
On the other hand, note that every element of V travels from χn to
ηn . Indeed, no dual curve in Dn has both endpoints on the same side of the
geodesic triangle. Hence any C ∈ V travels from χn to ηn since it cannot
cross νn and similarly any C ∈ H travels from χn to νn . Any geodesic joining
x to some point of νn must cross each element of V exactly once, and thus
each element of V occurs as a dual curve emanating from χn and terminating
on ηn , i.e. as an element of V. The same argument holds for H and H, and
thus the desired correspondences between dual curves and hyperplanes are
bijections.
Moreover, |V|, |H| ≥ n, since the distance from x to ηn , νn is precisely the
number of hyperplanes separating x from ηn , νn . Thus |V|, |H| ≥ n.
Crossing dual curves in Dn : Consider the decomposition χn =
c1 c2 . . . cm , where each ci is a 1-cube, with c1 initial and cm terminal. Suppose
x ∈ cp . Then each element of V is dual to ci with i ≤ p and each element of H
is dual to ci with i ≥ p. The dual curve emanating from cp belongs to V, H or
neither, according to the position of x on cp . Since the elements of V end on
ηn and the elements of H end on νn , each element of V crosses each element of
H and hence V and H are the two classes of a complete bipartite subgraph of
∆X.
Remark 4.1.10. A more detailed argument shows that, if X has finite degree,
then it is hyperbolic only if ΓX has thin bicliques. Roughly, large bicliques in
ΓX either yield large bicliques in ∆X or large cliques in ΓX. The only proof I
know is more long-winded than the utility of this fact warrants.
71
4.2
Hyperbolicity relative to hyperplane-stabilizers
This section is devoted to characterizing cocompactly cubulated groups
that are (strongly) hyperbolic relative to the collection of hyperplane stabilizers. Bowditch introduced the following convenient definition of relative
hyperbolicity:
Definition 4.2.1 (Relative hyperbolicity, fine graph [Bow97]). Let G be a
group an {Hi }i∈I a finite collection of finitely generated subgroups. Then
the pair (G, {Hi }) is relatively hyperbolic (or, G is hyperbolic relative to the
peripheral subgroups {Hi }) if G acts on a hyperbolic graph Γ such that:
1. There are finitely many orbits of edges in Γ and each edge-stabilizer is
finite.
2. The stabilizer of each vertex is either finite or conjugate to some Hi .
3. Each Hi stabilizes some vertex of Γ.
4. Γ is fine.
As usual, a cycle in a graph Γ is a combinatorial embedding S → Γ, where S
is some graph homeomorphic to a circle. The graph Γ is fine if for each edge
e of Γ and each n ∈ N, there are finitely many cycles of length n in Γ that
contain e.
The simplicity of this definition makes it a powerful tool for studying
relative hyperbolicity. The fine hyperbolic G-graph version of relative hyperbolicity is used, for example, in [MPW11] and in recent work of Bigdely-Wise,
to study relative quasiconvexity. Definition 4.2.1 was shown by Hruska to be
equivalent to Gromov’s definition and to the definition given by Osin using
relative presentations [Osi06, Hru10].
In our situation, G is a group acting properly and cocompactly on
the CAT(0) cube complex X, and S = {GW } is a collection of subgroups
containing one hyperplane-stabilizer from each conjugacy class. Note that
each GW is finitely generated, since it acts cocompactly on the contractible
space W . Moreover, by cocompactness, S is finite. G acts on the contact
graph ΓX in such a way that the set of vertex stabilizers is precisely the set
72
of conjugates of the GW . Moreover, since G acts cocompactly on X, there are
finitely many G-orbits of vertices and edges in ΓX, by Proposition 2.4.2. By
Theorem 3.1.1, ΓX is hyperbolic. It remains to give conditions under which
ΓX is fine and under which edge-stabilizers are finite. These turn out to be
closely related.
Definition 4.2.2 (Almost malnormal collection). Let G be a group and S a
collection of subgroups. S is an almost malnormal collection if, for any g ∈ G
and S1 , S2 ∈ S, either gS1 g −1 ∩ S2 is finite, or S1 = S2 and g ∈ S2 . If “finite”
can be replaced by “trivial”, the collection S is malnormal.
If ΓX is to be a G-graph satisfying Definition 4.2.1, then S must be
an almost-malnormal collection. Indeed, if H ⌣
⊥ H ′ is an edge of ΓX, then
GH⌣
⊥H ′ contains GH ∩ GH ′ as a subgroup of index at most 2 and therefore G
acts on ΓX with finite edge-stabilizers only if S is almost malnormal. This
is true of any peripheral structure in a relatively hyperbolic group, by the
same argument. Using the results of the next two sections, we prove the main
theorem:
Theorem 4.2.3. Let G act properly, cocompactly, and unambiguously on the
CAT(0) cube complex X. Then (G, S) is relatively hyperbolic if and only if S
is a malnormal collection. Hence, if G is hyperbolic relative to the collection of
hyperplane-stabilizers, then G is word-hyperbolic.
Proof. If (G, S) is relatively hyperbolic, then S must be almost malnormal, by
Definition 4.2.1.(1).
Since G acts unambiguously on X, the set {gSg −1 }g∈G is the set of
hyperplane stabilizers, and this set corresponds bijectively to the set of
hyperplanes. By Theorem 4.2.9, ΓX does not contain K2,∞ , and hence ΓX is
fine by Theorem 4.2.5. If G is not word-hyperbolic, then by Theorem 4.1.3,
there is an embedding of K∞,∞ in ∆X, and thus in ΓX, contradicting the fact
that ΓX does not contain K2,∞ . Hence G is word-hyperbolic.
73
Bowditch showed that a word-hyperbolic group is hyperbolic relative to
any finite almost malnormal collection of quasiconvex subgroups [Bow97],
and Theorem 4.2.3 is an example of this phenomenon since hyperplanes are
convex.
4.2.1
Fine contact graphs: the tiny bicliques property
Definition 4.2.4 (Tiny bicliques). The graph Γ has tiny bicliques if for all
embeddings Kp,q → Γ with p, q > 1, we have max{p, q} < ∞.
The goal of this section is to characterize CAT(0) cube complexes whose
contact graphs are fine, i.e. to prove:
Theorem 4.2.5. Let X be a CAT(0) cube complex of degree D < ∞. Then
ΓX is fine if and only if ΓX has tiny bicliques.
Proof. First suppose that ΓX does not have tiny bicliques. Then K2,∞ ⊆ ΓX,
and thus ΓX contains an edge H ⌣
⊥ H ′ that lies in infinitely many 4-cycles.
ΓX is therefore not fine.
Conversely, suppose that ΓX is not fine. Then, by Lemma 4.2.6, there
exists an edge f = V0 ⌣
⊥ V1 of ΓX and an infinite family {ρi }i∈Z of cycles such
that for all i ∈ Z, the cycle ρi contains f and either |ρi | = 3 for all i ∈ Z or
|ρi | = 4 for all i ∈ Z.
If each ρi is a 3-cycle, then let ρi = V0 ⌣
⊥ Wi ⌣
⊥ V1 ⌣
⊥ V0 . Since {Wi }i∈Z
is infinite, we have K({V0 , V1 }, {Wi }) ∼
= K2,∞ ⊆ ΓX, so that ΓX does not
have tiny bicliques. Hence suppose that |ρi | = 4 for each i ≥ 0. We shall
construct an embedding K2,∞ ,→ ΓX and thus show that ΓX does not have
tiny bicliques.
⊥ V1 ⌣
⊥ V0 . Without loss of
⊥ W1i ⌣
For each i ∈ Z, let ρi = V0 ⌣
⊥ W0i ⌣
generality, we may suppose that all of the hyperplanes in {W0i }i∈Z are distinct.
i
By local finiteness of X, the set {V1 , W0i }i contains an infinite subset {W0j }j≥0
such that the corresponding halfspaces are nested, i.e.
(W0i0 )+ ⊃ (W0i1 )+ ⊃ . . .
74
Indeed, the alternative is an infinite sequence of pairwise-contacting hyperplanes, contradicting local finiteness. Now for each j ≥ 0, the hyperplane V0
i
i
i
contacts W0j , which is separated from W0i0 by W0j−1 . Hence V0 ⊥W0j for all
j ≥ 0.
Figure 4–3: A heuristic picture of how 4-cycles give rise to complete bipartite subgraphs of ΓX. At least one of the two colored sets of hyperplanes is
infinite, and we thus obtain K2,∞ in ΓX.
i
i
If V1 ⊥W0j for all j ≥ 0, then K({V0 , V1 }, {W0j }) ∼
= K2,∞ ⊆ ΓX and we
are done. Otherwise, by removing finitely many of the W0ji , we have that
V1+ ⊃ (W0i0 )+ ⊃ (W0i1 )+ ⊃ . . .
i
Either V0 and some W1k both contact infinitely many hyperplanes W0j , so that
i
K({W1k , V0 }, {W0j }) ∼
= K2,∞ ⊆ ΓX, or there are infinitely many distinct W1k
and we have K({W 1 , V1 }, {W i }) ∼
= K2,∞ ⊆ ΓX. See Figure 4–3.
0
1
Lemma 4.2.6. Let e = V1 ⌣
⊥ V0 be an edge of ΓX that is contained in each
of the cycles γi , where {γi }i∈Z is an infinite family of distinct n-cycles and
n ≥ 3. Then there exists an edge f of ΓX and an infinite family {ρi }i∈Z of
distinct cycles such that each ρi has length 3 or each ρi has length 4, and f is
contained in each ρi .
Proof. By Lemma 4.2.8 below, for each i ∈ Z, there is a simple cycle γi′
to which γi e-reduces with |γi′ | ≤ 4. Assume that γi′ is optimal in the
following sense: if γi e-reduces to a 4-cycle, then γi′ is a simple 4-cycle to
75
which γi reduces. Otherwise, γi′ is the 3-cycle whose existence is guaranteed by
Lemma 4.2.8.
i
For each i ∈ Z, let γi = V0 ⌣
⊥ V1 ⌣
⊥ V2i ⌣
⊥ ... ⌣
⊥ Vn−1
⌣
⊥ V0 . By passing
if necessary to an infinite subsequence, we may assume, for all i, j ∈ Z, that
|γi′ | = |γj′ |. The 3- and 4-cycle cases are considered separately.
Strategy: In either case, the claim is proved if the set of γi′ actually
contains infinitely many distinct simple cycles. We therefore suppose that γi′
takes finitely many distinct values in the set of simple cycles as i varies, and
use this assumption to locate an edge e1 of Γ and an infinite family of simple
cycles, each of length k < n and containing e1 . The lemma then follows by
induction on the length n.
The 3-cycle case: Let Wi be the third hyperplane of the 3-cycle γi′ ,
so that γi′ = V0 ⌣
⊥ Wi ⌣
⊥ V1 ⌣
⊥ V0 . By hypothesis, there are finitely many
distinct hyperplanes Wi , so, applying the pigeonhole principle and passing
to an infinite subsequence, we may suppose that for all i ∈ Z, the cycle
γi′ = γ = V0 ⌣
⊥ W ⌣
⊥ V1 for some hyperplane W . The cycle γi is of one of
three cycle species with respect to γ. A Species I cycle γi is one in which W
corresponds to a vertex of γi , so that γ 0 ⊂ γi0 . The cycle γi is of Species II
if W contacts Vki with k = n − 1 or k = 2 and does not belong to γi . The
cycle γi is of Species III if W contacts Vki with 3 ≤ k ≤ n − 2 and neither
i
contacts Vn−1
or V2i nor belongs to γi . Since γi e-reduced to γ, these are the
only possibilities. See Figure 4–4.
Figure 4–4: The three species of 3-cycle.
76
For a ∈ {I, II, III}, let Ja ⊂ Z be the set of i such that γi is of Species a
with respect to γ. Then at least one of the sets Ja is infinite, so without loss
of generality, it suffices to suppose successively that Ja = Z for each possible
value of a.
Species I: Let e1 = W ⌣
⊥ V1 and let γ̂i be the subpath of γi joining V1
to W that does not contain V0 and let γ̆i be the subpath joining W to V1 and
containing V0 . Then each of e1 γ̂i and e1 γ̆i is a simple cycle of length at most
n − 1. Since {γi }i is infinite, so is {e1 γ̂i }i ∪ {ei γ̆i }i . Thus there is an infinite
family of simple cycles of length at most n − 1, each containing e1 .
Species II: If γi is of Species II with respect to γ, then V0 ⌣V
⊥ 1 ⌣W
⊥ ⌣V
⊥ n−1 ⌣V
⊥ 0
is a 4-cycle to which γi e-reduces, contradicting the assumption that γi′ is optimal.
Species III: Let e1 = V1 ⌣
⊥ W , let f = W ⌣
⊥ Vki , and let γ̂i be the subpath
of γi joining Vki and V1 and not containing V0 . Let γ̆i be the closure in γi of
γi \ γ̂i . Then each of e1 f γ̂i and e1 f γ̆i is a simple cycle of length at most n − 1
containing e1 . We thus argue as in Species I and conclude that there is an
infinite family of simple cycles of length at most n − 1 containing e1 .
The 4-cycle case: We may suppose that all of the γi reduce to the same
4-cycle γ = V0 ⌣
⊥ V1 ⌣
⊥ W1 ⌣
⊥ W0 ⌣
⊥ V0 . For each i ∈ Z, the cycle γi has a flavor
with respect to γ. The cycle γi is of Flavor I if W0 and W1 do not belong to
γi , and W0 ⌣
⊥ Vki and W1 ⌣
⊥ Vli with 2 ≤ k, l ≤ n − 1. A Flavor II cycle γi is
one in which W1 (say) is contained in γi and W0 is not. A Flavor III cycle γi
is one in which both W0 and W1 are in γi .
Flavor I: Let Vki ⌣
⊥ W0 and Vli ⌣
⊥ W1 , with k ̸= 0 and l ̸= 1. For each
Flavor I γi , let αi be the simple subpath of γi joining Vki to V0 that does not
contain V0 and βi the analogous path for Vli and V1 , so that αi eβi is a subpath
of γi . If αi ∩ βi contains at least one edge, then |αi | ≤ n − 3, as at left in
Figure 4–6. Let fi = W0 ⌣
⊥ Vki and let e1 = V0 ⌣
⊥ W0 . Then e1 f αi is a simple
cycle of length at most n − 1 containing e1 .
77
Figure 4–5: The three flavors of 4-cycle.
Figure 4–6: Flavor I possibilities.
Otherwise, the closure in γi of the complement of αi ∪ βi contains a
path γ̄i , disjoint from e, of length n − |αi | − |βi | − 1. Let e1 = W0 ⌣
⊥ W1
and f01 = W0 ⌣
⊥ Vki and f2i = W1 ⌣
⊥ Vli . Then f0i γ̄i e1 f1i is a simple cycle
of length |γ̄i | + 3 containing e1 . Hence either there is a cycle of length less
than n containing e1 or |αi | = |βi | = 1, as at right in Figure 4–6. In the
i
latter case, we may assume that Vn−1
and V2i are fixed as i varies. Indeed, if
i
there are infinitely many different values of Vn−1
as i varies, then there are
i
infinitely many 3-cycles of the form V0 ⌣
⊥ W0 ⌣
⊥ Vn−1
⌣
⊥ V0 and the lemma
is proved. Hence, as usual, we may apply the pigeonhole principle and pass
to a suitable infinite subsequence, so that γi is determined by the path γ̄i .
i
Thus let e1 = W0 ⌣
⊥ W1 , let f0 = W0 ⌣
⊥ Vn−1
and let f1 = W1 ⌣
⊥ V2i . Then
{f0 e1 f1 γ̄i }i is an infinite family of length-n simple cycles containing the path
f0 e1 f1 . Proceeding inductively, we find after a finite number of applications
of this argument that either γi = γj for all i, j, a contradiction, or there is
some edge e′ and a family of n-cycles containing e′ , each of which reduces to a
3-cycle or a 4-cycle with respect to which it has Flavor II or III.
78
Flavor II: Let e1 = W0 ⌣
⊥ W1 and let f = W0 ⌣
⊥ V i , where V i is a
hyperplane in γi , different from V0 and V1 . For each i such that γi is of Flavor
II, let γ̄i be the simple subpath of γi joining V i to W1 and not containing V0 .
Observing that |γ̄i | ≤ n−2 shows that e1 f γi is a simple cycle of length at most
n − 1 that contains e1 . Hence if there are infinitely many Flavor II cycles,
there are infinitely many cycles of length strictly less than n that contain the
edge e1 .
Flavor III: This is the simplest case. Let e1 = W0 ⌣
⊥ W1 and for each
Flavor III cycle γi , let γ̄i be the simple subpath of γi joining W0 to W1 and
not containing e. Then the simple cycle e1 γ̄i has length n − 2 and contains e1 .
Hence if there are infinitely many Flavor III cycles, there are infinitely many
(n − 2)-cycles that contain the edge e1 .
Induction: The above argument shows that there is an infinite family
{γi1 }i of simple cycles containing the edge e1 , with |γi1 | < n for all i. By the
pigeonhole principle, infinitely many of the γi1 have the same length, so we
may assume that |γi1 | = k < n for all i ∈ Z. Hence the above argument may be
applied as long as k > 4, and the lemma follows by induction.
The following definition and lemma support the proof of Lemma 4.2.6.
Definition 4.2.7 (Condensed cycle, semi-condensed cycle, cycle-reduction).
Let γ = W0 ⌣
⊥ W1 ⌣
⊥ W2 ⌣
⊥ W3 ⌣
⊥ W0 be a 4-cycle in ΓX. Then γ is
semicondensed if W0 ⌣
⊥ W2 or W1 ⌣
⊥ W3 and condensed if W0 ⌣
⊥ W2 and
W1 ⌣W
⊥ 3 . See Figure 4–7.
Let γ ⊂ ΓX be a simple cycle containing the edge e with |γ| = n. The
cycle γ e-reduces to the simple cycle γ ′ if all of the following hold: (1) γ ′
contains e, and (2) |γ ′ | ≤ |γ|, and (3) every vertex of γ ′ is either contained in
γ or is adjacent to a vertex of γ − e.
Cycles can be reduced, using disc diagrams, in a simple way.
79
Figure 4–7: The 4-cycle γ is neither semicondensed nor condensed at left,
semicondensed in the center, and condensed at right. The corresponding full
subgraphs of Γ are shown below. There are other possibilities, according to
whether the illustrated edges of Γ correspond to crossings or osculations.
Warning: while failure to be (semi)condensed is always reflected in some disc
diagram as shown, the converse is not true, since a separating dual curve may
map to a hyperplane that crosses one of the labeled hyperplanes outside of the
disc diagram.
Lemma 4.2.8. Let γ → ΓX be a simple cycle of length n > 4 containing the
edge e. Then γ e-reduces to a cycle ρ with |ρ| ≤ 4. If |γ| = 4, then γ e-reduces
to a condensed 4-cycle ρ.
Proof. Let γ = V0 ⌣
⊥ V1 ⌣
⊥ ... ⌣
⊥ Vn−1 ⌣
⊥ V0 and, for each i, let σi → N (Vi )
be a combinatorial geodesic segment such that σi ends on the initial 0-cube
e
of σi+1 for all i ∈ Zn , and let e = V0 ⌣
⊥ V1 . There is a disc diagram D → X
∏
of minimal area among diagrams with boundary path i σi . If |σ0 | = 0, then
V0 ⌣
⊥ V1 ⌣
⊥ Vn−1 ⌣
⊥ V0 is a 3-cycle to which γ e-reduces. If |σ0 | > 0, then there
is a 1-cube c of σ0 that has a 0-cube belonging to σ1 but is not dual to V1 , so
that there is a dual curve C in D emanating from c. The dual curve C maps
to a hyperplane W that contacts V0 and V1 , so that γ reduces to the 3-cycle
V0 ⌣
⊥ V1 ⌣
⊥W⌣
⊥ V0 , as in Figure 4–8. Note that this is indeed a 3-cycle, since
W ̸= V0 and W ̸= V1 , for otherwise V0 and V1 would self-contact, which is
impossible in CAT(0) cube complexes.
80
Figure 4–8: e-reducing to a 3-cycle.
If γ is a 4-cycle, then every hyperplane H separating V0 from V2 must
cross every hyperplane U separating V1 from V3 , and we obtain a 4-cycle
V0 ⌣
⊥ V1 ⌣
⊥H⌣
⊥U⌣
⊥ V0 . The hyperplanes U and V appear as dual curves in
D, and hence, by choosing a 4-cycle containing V0 ⌣
⊥ V1 in such a way that the
area of the resulting diagram D is minimal, we therefore choose a condensed
4-cycle.
4.2.2
Almost malnormality of hyperplane stabilizers
It remains to show that, if G acts unambiguously, properly, and cocompactly on X, and S is an almost malnormal collection, then ΓX is fine. This
happens in the next theorem, for which, in the interest of contrasting median
graph and cube complex methodology, we give two proofs; one uses media,
and the other is by induction on dimension, invoking the fact that each hyperplane in a D-dimensional CAT(0) cube complex is itself a CAT(0) cube
complex that is at most (D − 1)-dimensional [Sag95].
Theorem 4.2.9. Let G act properly, cocompactly, and unambiguously on X.
If S is an almost malnormal collection, then K2,∞ ̸,→ ΓX.
Proof. Suppose that K2,∞ ,→ ΓX, so that there exists a pair of distinct
hyperplanes U0 , U1 and an infinite family {Wi }i∈Z such that for all i ∈ Z, we
81
have U0 ⌣W
⊥ i ⌣U
⊥ 1 . Note that if U is a hyperplane separating U0 from U1 , then
U ⊥Wi for each i ∈ Z, and hence we suppose that U0 ⌣U
⊥ 1.
Since G acts properly and cocompactly, X is locally finite. Hence there
exists an infinite set I ⊂ Z such that, if i < j < k all belong to I, then Wj
separates Wi from Wk . Hence there exists an infinite subset I ′ ⊂ I such that
U0 ⊥Wi ⊥U1 for all i ∈ I ′ .
Since G acts with finitely many orbits of hyperplanes, there exists
W ∈ {Wi }i∈I ′ and an infinite set hj ∈ G such that hj W ̸= hk W for all i ̸= k
and for each j, we have hj W = Wi for some i ∈ I ′ . Thus, by renumbering
things, we have h0 , h1 , . . . ∈ G such that U0 ⊥hi W ⊥U1 for all i ≥ 0 and,
whenever 0 ≤ i < j < k, the hyperplanes hi W and hk W are separated by
hj W . See the left side of Figure 4–9. There are now two ways to complete the
argument, one by median techniques and one by cubical techniques.
Figure 4–9: At left is the situation arising from K2,∞ in the contact graph of
a cocompact cubulation. At right is the median argument showing that this
situation contradicts malnormality of the hyperplanes.
The median approach: By Helly’s theorem, we can choose a 0cube x0 ∈ N (U0 ) ∩ N (U1 ) ∩ N (W ). For each i ≥ 0, choose a 0-cube
xi ∈ N (U0 ) that is separated from x0 by hi W . For example, choose any 0-cube
in N (hi+1 W ) ∩ N (U0 ). Likewise, choose a 0-cube yi ∈ N (U1 ) that is separated
from x0 by Wi .
Let mi = m(x0 , xi , yi ) be the median of these three 0-cubes. Then mi is
separated from x0 by Wi , since the geodesic segment that joins xi to yi and
82
passes through mi lies in the halfspace associated to Wi that contains xi , yi , by
convexity of halfspaces.
Let Pi be the geodesic segment joining x0 to mi that extends to geodesic
segments joining x0 to xi and to yi ; see the right side of Figure 4–9. Since
N (U0 ) and N (U1 ) are convex, Pi ⊂ N (U0 ) ∩ N (U1 ).
Hence N (U0 ) ∩ N (U1 ) has infinite diameter. Indeed, mi ∈ N (U0 ) ∩ N (U1 )
and dX (x0 , mi ) = |Pi | ≥ i, since Wj separates mi from W, and thus from x0 ,
for all j ≤ i. Since N (U0 ) ∩ N (U1 ) is convex, its stabilizer acts cocompactly,
and thus GN (U0 )∩N (U1 ) is infinite. This implies that GU0 ∩ GU1 is infinite.
But, since G acts unambiguously on X and U0 ̸= U1 , this intersection is the
intersection of distinct conjugates of elements of S, which is therefore not an
almost-malnormal collection.
Induction on dimension: By cocompactness, dim X = D < ∞, and we
shall argue by induction on D. If D = 0, then X is a single 0-cube; since there
are no hyperplanes, for all K2,∞ in the (empty) contact graph, there exists a
pair of hyperplane-stabilizers with infinite intersection.
For D ≥ 1, we are in the same situation as shown at left in Figure 4–9:
we have U0 ⌣
⊥ U1 and an infinite collection of elements hj ∈ G such that
U0 ⊥hj W ⊥U1 for all j ≥ 0 and, whenever 0 ≤ i < j < k, the hyperplanes hi W
and hk W are separated by hk W .
We may assume that each hj ∈ GU0 . Indeed, choose a 2-cube s0 ⊂
N (U0 ) ∩ N (W ), two of whose 1-cubes are dual to U0 and two of whose 1-cubes
are dual to W . On the other hand, there is a 2-cube si0 ⊂ N (U0 ) ∩ hi N (W ),
two of whose 1-cubes are dual to U0 and two of whose 1-cubes are dual to
hi W . Since there are finitely many orbits of 2-cubes, and infinitely many such
2-cubes si0 , corresponding to crossings U0 ⊥hi W , there exist infinitely many si0
that belong to the same orbit, i.e. we can choose W and s0 ⊂ N (U0 ) ∩ N (W )
so that there exist infinitely many ki ∈ G, with each ki W = hi W , such
83
that si0 = ki s0 . Hence each ki ∈ GU0 . To see this, choose a base 0-cube
x0 ∈ s0 . Then there is a geodesic segment Qi → N (U0 ) joining x0 to
ki x0 ∈ ki s0 ⊂ N (U0 ). Under the projection X → X, the path Qi projects to a
closed based path in the carrier of the immersed hyperplane U0 representing
the element ki . Thus we can assume that each hi ∈ GU0 .
Fix a base 0-cube y0 ∈ N (U0 ) ∩ N (U1 ) ∩ N (h0 W ). Then hj U1 contacts
U0 at hj y0 for each j ≥ 0. Suppose that there exists h ∈ GU0 such that
hj U1 = hU1 for infinitely many values of j. There are therefore infinitely many
hj ∈ GU0 such that hj U1 = hU1 . Hence h−1 hj ∈ GU1 ∩ hGU0 = GU1 ∩ GU0 .
Since U0 ̸= U1 , minimality of the action of G on X implies that the collection
of hyperplane-stabilizers is not almost malnormal, and thus neither is S.
If this is not the case then, by passing to an infinite subsequence, we may
assume that hj U1 ̸= hk U1 for j ̸= k. Now for all j ≥ 0, we have hj U1 ⊥hj W ,
since U1 ⊥W . On the other hand, we also have U1 ⊥hj W , by hypothesis, and
also U0 ⌣
⊥ hj U1 since hj U0 ⌣
⊥ hj U1 and hj ∈ GU0 . By the same local finiteness
argument as was used at the beginning of the proof, we have that hj U1 ⊥U0 for
infinitely many j.
We now work in the cube complex U0 , which has dimension strictly less
than D. Note that GU0 acts properly and cocompactly on U0 . Let h0 U1 , h1 U1
cross U0 , so that U0 contains hyperplanes H0 = h0 U1 , H1 = h1 U1 and
Vj = hj W . By Helly’s theorem and convexity of hyperplanes, for all j ≥ 0,
we have H0 ⊥Vj ⊥H1 . By induction on dimension, StabU0 (H0 ) ∩ StabU0 (H1 ) is
infinite, from which it follows that GU0 ∩ Gh0 U1 is infinite, and thus GU0 ∩ GU1
is infinite, since h0 ∈ GU0 . Thus S is not almost malnormal.
4.3
Fractional flats and their crossing graphs
We conclude this chapter with a very concrete discussion of isometrically
embedded subcomplexes of R2 , the fractional flats, and their crossing graphs.
Fractional flats are closely related to the thin bicliques condition, and play a
large role in Chapter 5 and in Chapter 6.
84
4.3.1
Fractional flats
Recall that R is the standard cubulation of Z, i.e. R is a CAT(0) cube
complex homeomorphic to R, with 0-cubes at half-integer points. Likewise,
R+ is a combinatorial ray. A (cubical) flat is R2 , i.e. the standard tiling of R2
by 2-cubes. A non-diagonal half-flat is R × R+ . A non-diagonal quarter-flat is
R+ 2 . See Figure 4–10.
Figure 4–10: Clockwise from left: a flat, a nondiagonal half-flat, and a nondiagonal quarter-flat.
Note that the hyperplanes in R2 come in two ordered families, {Vi }i∈Z
and {Hi }i∈Z , such that for all i ∈ Z, the hyperplane Hi separates Hi+1 from
Hi−1 , the hyperplane Vi separates Vi−1 from Vi+1 , and Vi ⊥Hj for all i, j ∈ Z.
The automorphism group of R2 is Z2 o D4 . The automorphism group
of a non-diagonal half-flat is virtually Z, and the automorphism group of a
non-diagonal quarter-flat is finite.
An eighth-flat E is an isometrically embedded subcomplex isomorphic
to any cube complex formed as follows. Let γ be a combinatorial geodesic
ray in R+ 2 whose 1-cubes are dual to the hyperplanes {Vi }i≥0 . Let δ be any
combinatorial geodesic ray in R+ 2 containing a 1-cube dual to each of the
hyperplanes in {Vi }i≥0 ∪ {Hj }j≥0 . The subcomplex bounded by γ and δ,
consisting of 2-cubes dual to hyperplanes in the above-named sets, is E. See
Figure 4–11.
85
Figure 4–11: An eighth-flat.
Note that the automorphism group of an eighth-flat is trivial. The other
salient feature of E is that, while there is a geodesic ray, namely γ, dual to
the set of “vertical” hyperplanes, and a geodesic ray, namely δ, dual to all of
the hyperplanes, there is no geodesic ray dual to exactly the set of horizontal
hyperplanes. This feature of eighth-flats is important in our later discussion of
visible simplices at infinity.
Let α be a combinatorial geodesic ray with W(α) = {Vi }i≥0 ∪ {Hj }j≤0
and let β be a combinatorial geodesic ray with W(β) = {Vi }i<0 ∪ {Hj }j≥0 ,
and let Q be the isometrically embedded subspace of R2 bounded by α and β.
Any such Q is a diagonal quarter-flat. Equivalently, let E1 , E2 be eighth-flats,
and let Q be formed by gluing E1 to E2 along the “straight” bounding rays
that cross only vertical hyperplanes. Then Q is a diagonal quarter-flat. See
Figure 4–12.
Figure 4–12: A diagonal quarter-flat.
Finally, let σ be a bi-infinite combinatorial geodesic in R2 that crosses
each hyperplane exactly once. The closure of either of the components of
R2 −σ is a diagonal half-flat. Note that a diagonal half-flat F may have finite
86
automorphism group, or Aut(F) may be virtually Z, depending on whether σ
is a periodic geodesic in R2 .
A fractional flat in X is an isometrically embedded subcomplex F ⊆ X
such that F is isomorphic to a flat, half-flat, quarter-flat, eighth-flat, diagonal
quarter-flat, or diagonal half-flat. Note that a fractional flat does not have
thin bicliques or tiny bicliques, since ∆F contains Kp,q with min(p, q) arbitrarily large, and ∆F contains K2,∞ . For each of the nondiagonal fractional flats
F, there is a copy of KN,N in ∆F.
We emphasize that we are using the word flat in a more specific way than
the general usage, since X may contain subspaces isometric to R2 that are
not cubical flats in our sense. However, in a cocompact setting, Theorem 4.1.3
tells us that failure to be hyperbolic guarantees that a CAT(0) cube complex
contains a genuine cubical flat.
4.3.2
Nearjoins: crossing graphs of fractional flats
The following motivates the notion of a unidirectional boundary set
defined in Chapter 5.
Let V = {Vi }i≥1 and H = {Hj }j≥1 be sets of vertices. Let I : N → N be
a nondecreasing function. Then the weak nearjoin WN(H, V, I) is the graph
whose vertex set is V ⊔ H, with an edge joining each Hj to Vi for all i ≥ I(j).
If I is bounded, then Hj is adjacent to Vi for all sufficiently large i, j, and
WN(H, V, I) contains KN,N as a cofinite subgraph. Otherwise, WN(H, V, I)
contains K2,∞ , as well as Kp,p for all p ≥ 0, but does not contain KN,N . In
this case, WN(H, V, I) is a proper weak nearjoin. Note that, in a proper weak
nearjoin, each Hj has infinite valence, and each Vi has finite valence. If E
is an eighth-flat, then, with respect to the obvious orderings on the vertical
and horizontal hyperplanes, there exists I such that ∆E ∼
= WN(H, V, I)
is a proper weak nearjoin. See Figure 4–13. Note that a connected weak
nearjoin WN(V, H, I) has diameter at most 3, since every Vi is adjacent to
H1 . Hence the contact graph of an eighth-flat E has finite diameter (although
87
Figure 4–13: Part of a proper weak nearjoin, arising as the crossing-graph of
an eighth-flat.
the crossing graph may contain finitely many isolated vertices, if the two
bounding rays of E have a common initial segment of length at least 1).
Now let V = {Vi }i∈Z and H = {Hj }j∈Z . The nearjoin of V and H,
denoted N(V, H, I, J), is a graph with vertex set V ⊔ H arising from a pair of
functions I : Z → Z ∪ {∞}, J : Z → Z ∪ {−∞} such that I is non-decreasing, J
is non-increasing, and Vi is adjacent to Hj for all j ≥ J(i) and Hj is adjacent
to Vi for all i ≤ I(j). If I, J take only finite values, then N(V, H, I, J) is a
proper nearjoin. For example, the crossing graph of a diagonal half-flat is a
proper nearjoin.
We always use these terms in the following context. Let V = {Vi } and
H = {Hj } be sets of hyperplanes so that no two halfspaces in {Vi+ } are
disjoint, and no two halfspace in {Hj+ } are disjoint. Then each of these sets
of halfspaces is partially ordered by inclusion; indices are assigned so that if,
for example, Vj separates Vi , Vk , then i < j < k or k < j < i. We require that
V, H not contain any facing triple of hyperplanes, i.e. that Vi+ , Vj+ , Vk+ are
pairwise-comparable in the partial order if Vi , Vj , Vk are pairwise non-crossing.
Then V ⊔ H generates a weak nearjoin in ∆X if each Hj crosses all but finitely
many Vi .
88
CHAPTER 5
The simplicial boundary of a cube complex
In this chapter, we define the simplicial boundary of X, denoted ∂△ X,
which is a “simplicial complex at infinity” isolating the non-hyperbolic
behavior of X. In Section 5.1, we introduce unidirectional boundary sets
and prove a structure theorem allowing us to define ∂△ X and state some of
its essential properties. In Section 5.2, we introduce the notion of a “visible
simplex at infinity” and discuss the extent to which ∂△ X can be viewed as a
“space of rays” in X1 .
5.1
Definition and basic properties
5.1.1
Unidirectional boundary sets
Throughout this chapter, X is strongly locally finite, in the sense that
there is no infinite family of pairwise-contacting hyperplanes. This
condition is required to define the simplicial boundary, and is described in
greater detail in Section 1 of [Hag12].
As in [SW05, CS11], the hyperplanes U, V, W ∈ W form a facing triple
if there are halfspaces U + , V + , W + such that U, V ⊂ W + , U, W ⊂ V + , and
V, W ⊂ U + . In other words, U, V, W are pairwise non-crossing, and no set of
three halfspaces, with one associated to each of these hyperplanes, is totally
ordered by inclusion.
Lemma 5.1.1. If U ′ is an infinite set of hyperplanes, then there is an infinite
set U ′′ ⊂ U ′ of pairwise non-crossing hyperplanes containing no facing triple.
Proof. From strong local finiteness, it follows that there exists {Ui }i≥0 ⊂ U ′
such that Ui separates Ui−1 from Ui+1 for all i ≥ 1. Let U ′′ = {Ui }i≥0 . If
i < j < k and Ui , Uj , Uk form a facing triple, then Uj does not separate Ui
from Uk , a contradiction.
89
To construct {Ui }, choose a hyperplane U0 . If U ⊥U0 for all U ∈ U ′ −{U0 },
then choose U0 differently: since X is locally finite and U ′ is infinite, it cannot
be the case that the elements of U ′ pairwise cross. Choose U1 ∈ U ′ − {U0 }
that does not cross U0 . By the same argument, either U0 separates U1 from
some U2 ∈ U ′ or vice versa. Indeed, the alternative is that U ′ contains an
infinite family of pairwise-osculating hyperplanes, which also contradicts local
finiteness. Continuing in this manner, we obtain the desired set.
The set U ⊂ W is nested if there is a 0-cube f (possibly at infinity) such
that {f (U ) : U ∈ U} is totally ordered by inclusion. The set U is semi-nested
if there is a 0-cube f at infinity such that {f (U ) : U ∈ U} is partially ordered
by inclusion and, if U ′ ⊂ U is a set of pairwise non-crossing hyperplanes, then
{f (U ) : U ∈ U ′ } is totally ordered by inclusion. The hyperplane U0 in the
semi-nested set U is initial with respect to f if for all U ∈ U − {U0 }, we do
not have U0 ⊂ f (U ), i.e. U either crosses U0 or U ⊂ f (U0 ). Note that, if U is
semi-nested, then U cannot contain a facing triple.
If f is a 0-cube at infinity, then f partially orders W: for hyperplanes
W1 , W2 , let W1 ≺ W2 if and only if f (W2 ) ⊂ f (W1 ). Lemma 5.1.2 gives
an orientation f of W that induces such a partial order on the semi-nested
set U in such a way that any two hyperplanes cross or are comparable. See
Figure 5–1.
Figure 5–1: No facing triple implies semi-nested.
90
Lemma 5.1.2. Let U be an infinite, inseparable set of hyperplanes containing
no facing triple. Suppose moreover that U is unidirectional in the sense
of Definition 5.1.3.(4). Then there exists a 0-cube f at infinity such that
the partial order ≺ on U induced by the inclusion-partial order on {f (U ) :
U ∈ U} has the property that any two hyperplanes in U either cross or are
≺-comparable.
Proof. Choose U0 ∈ U, and choose a halfspace f (U0 ). For each hyperplane
U ⊂ f (U0 ), let f (U ) be the halfspace associated to U that does not contain
U0 , so that f (U ) ⊂ f (U0 ). For every hyperplane V ⊂ X − f (U0 ), let f (V )
be the halfspace containing U0 , so that f (U0 ) ⊂ f (V ). This gives a consistent
orientation of all hyperplanes that do not cross U0 .
Let A be the set of hyperplanes in f (U0 ), i.e. those oriented away from
U0 by f . Let B be the set of hyperplanes in X − f (U0 ), i.e. those oriented
toward U0 by f . If W ⊥U0 , then there are four possibilities
1. W is of facing type if there exists A ∈ A and B ∈ B such that
W ⊂ f (A) ∩ X − f (B). In this case, f (W ) is the halfspace containing A
and B.
2. W is of A-facing type if W ⊂ f (A) for some A ∈ A, but W ⊥B for all
B ∈ B. In this case, f (W ) is the halfspace not containing A.
3. W is of B-facing type if W ⊂ X − f (B) for some B ∈ B, but W ⊥A for
all A ∈ A. In this case, f (W ) is the halfspace containing B.
4. W is of crossing type if W ⊥V for all V ∈ A ∪ B. In this case, any
choice of f (W ) is consistent with f |A∪B . If W is of crossing type, then
let f (W ) be the halfspace that has nonempty intersection with f (V ) for
all hyperplanes V of one of the facing types.
f is consistent: The map f consistently orients A, B by construction.
If A ∈ A and B ∈ B, then f (B) ⊂ f (A), whence f consistently orients
A ∪ B ∪ {U0 }. As shown in Figure 5–2, it is easily seen that W consistently
orients all hyperplanes of A ∪ B ∪ {U0 } together with the set of hyperplanes of
any of the three facing types. Hence f is consistent.
91
Figure 5–2: The map f consistently orients all hyperplanes of facing type.
f is non-canonical: If A or B is infinite, then f is non-canonical since f
differs on infinitely many hyperplanes from any x0 ∈ N (U0 ). Since X is locally
finite and U is infinite, we can choose U0 such that A ∩ U or B ∩ U is infinite,
by Lemma 5.1.1. Since U is unidirectional, we choose U0 so that A ∩ U = ∅
and B ∩ U is infinite.
f semi-nests U: Define a partial order ≺ on U by U ≺ V if f (V ) ⊂
f (U ). If U, V ∈ U do not cross and are ≺-incomparable, then f orients U
toward V and V toward U. Moreover, U must cross U0 , say, for otherwise
U0 , U, V would be a facing triple in U. If V does not cross U0 , then V if
oriented away from U0 and hence either V is oriented away from U , or U ⊥V, a
contradiction. Thus f semi-nests U.
This discussion motivates the notion of a unidirectional boundary set.
Definition 5.1.3 (Unidirectional boundary set, almost-equivalence). A
unidirectional boundary set (UBS) U is a set of hyperplanes satisfying:
1. U is infinite.
2. U is inseparable.
3. U contains no facing triple, i.e. U is semi-nested.
92
4. For each hyperplane U ∈ U, at most one of the associated halfspaces
contains infinitely many U ′ ∈ U. In particular, every semi-nested subset
of U has an initial hyperplane.
If U satisfies (1) − (3) but contains at least one semi-nested subset with
no initial hyperplane, then U is a bi-directional boundary set. The UBS U1
consumes the UBS U2 if all but finitely many H ∈ U2 lie in U1 . If U1 consumes
U2 and vice versa, then U1 and U2 are almost-equivalent. In other words, U1
and U2 are almost-equivalent if U1 △U2 is finite. The UBS U is minimal if
every UBS contained in U is almost-equivalent to U. See Figure 5–3 and
Figure 5–3: At left is a minimal UBS. At right is a non-minimal UBS.
Figure 5–4: The set of hyperplanes at left is bi-infinite. The set in the center
contains a facing triple. The set of un-arrowed hyperplanes at right is not
inseparable, since the arrowed hyperplane separates two of its hyperplanes.
Figure 5–4.
The motivating case is that of U = W(γ) for some combinatorial geodesic
ray γ. Accordingly, if γ, γ ′ : R+ → X are combinatorial geodesic rays, we say
that γ consumes γ ′ if W(γ) consumes W(γ ′ ), and that γ and γ ′ are almostequivalent if the corresponding UBSs are almost-equivalent. If W(γ ′ ) = W(γ),
then γ, γ ′ are hyperplane-equivalent. The requirement that a UBS not contain
a facing triple reflects that fact that a UBS is a generalization of the set of
hyperplanes dual to the 1-cubes of a combinatorial geodesic ray.
93
If U ′ is a set of hyperplanes, then the inseparable closure of U ′ is the
intersection of all inseparable sets U such that U ′ ⊆ U. Note that the
inseparable closure of U ′ consists of U, together with every hyperplane V ∈ W
for which there exist U1 , U2 ∈ U that are separated by V . Note also that if U ′
is contained in the inseparable set V, then so is the inseparable closure of U ′ .
Proposition 5.1.4. Let U ′ ⊆ W be an infinite, inseparable set of hyperplanes.
Then U ′ contains a minimal UBS. If U is a hyperplane in the leafless cube
complex X, then U is initial in some minimal UBS.
Proof. We verify three separate claims:
U ′ contains a UBS: Since X is locally finite and U ′ is infinite, there is
an infinite set U ′′ ⊆ U ′ of pairwise non-crossing hyperplanes. By Lemma 5.1.1,
there exists a nested set U ′′′ ⊂ U ′′ with one initial hyperplane U0 , such that
U ′′′ contains no facing triple. Let {Ui }i≥0 be an enumeration of U ′′′ , such that
for all i ≥ 1, the hyperplanes Ui+1 and Ui−1 are separated by Ui . Let U be
the inseparable closure of U ′′′ , which is contained in U ′ . Then U is infinite,
inseparable, and unidirectional. It suffices to verify that U contains no facing
triple.
Hyperplanes U1 , U2 , U3 ∈ U ′′′ cannot form a facing triple. Let V = U − U ′′′ .
Then any facing triple in U must contain at least one element of V. If, for
i < j, we have Ui , Uj ∈ U ′′′ and Ui , Uj , V is a facing triple, then Ui and Uj lie
in a single halfspace V + associated to V . Since U ′′′ is nested, UI , UJ ⊂ V + for
I ≤ i and J ≥ j. If i < k < j and Uk ⊂ V − , then Uk , Ui , Uj form a facing
triple, a contradiction. Thus V cannot separate any two elements of U ′′′ , and
hence V ̸∈ U.
If Ui , V1 , V2 form a facing triple and V1 , V2 ∈ V, then there exists Uj1
that is separated from Ui by V1 . This forces Ui , Uj1 , V2 to form a facing triple,
which is impossible by the preceding argument. An identical argument precludes facing triples of the form V1 , V2 , V3 in U. Hence U is a UBS containing
U ′.
94
Each UBS contains a minimal UBS: Let S ⊆ U be a UBS. If S does
not contain Ui , then S cannot contain any U ∈ U that separates Ui from U0 .
Thus there exists some minimal I < ∞ such that S contains Ui for i ≥ I. If
U ∈ U does not separate UI from U0 and does not cross Ui , then inseparability
of S requires that U ∈ S. Hence every element of U − S either separates U0
from UI or crosses UI and separates U0 from Ui for some i ≥ I.
Indeed, if U ∈ U, then U cannot cross all but finitely many of the Ui , for
otherwise U − {U } would be a smaller inseparable set containing U ′ . Let C1
be the set of hyperplanes in U that cross Ui . If C1 is finite, then U − S has
cardinality |C1 | + dX (N (U0 ), N (Ui )) and so S is almost-equivalent to U.
If C1 is infinite, then let U1 = U − S. Note that C1 is unidirectional and
contains no facing triple, since those properties are inherited by subsets. If
C, C ′ ∈ C1 are separated by some hyperplane W , then W crosses UI and
crosses each hyperplane Uj crossed by both C and C ′ . Moreover, since W
does not cross C or C ′ , W does not cross Uj for j sufficiently large. Hence
W ∈ C1 , which is therefore a unidirectional boundary set. Likewise, U1 is
infinite, unidirectional, and contains no facing triple. By the definition of C1 ,
U1 is also inseparable. Hence U1 is a unidirectional boundary set. Moreover,
every element of U1 crosses all but finitely many elements of C1 .
Hence, if U is not minimal, then U contains two disjoint unidirectional
boundary sets, U1 and C1 , such that every hyperplane in U1 crosses all but
finitely many hyperplanes in C1 . If U1 is not minimal, then by the same
reasoning, U1 is almost-equivalent to a disjoint union U2 ⊔ C2 of unidirectional
boundary sets such that each U ∈ U2 crosses all but finitely many C ∈ C2 ,
and each of U and C crosses all but finitely many C ′ ∈ C1 . Apply the same
argument to U2 , and continue in this way. Since X does not contain an infinite
set of pairwise-crossing hyperplanes, we must, after finitely many applications
of this argument, find a minimal unidirectional boundary set contained in U.
The non-leaf U is initial in a UBS: Let U0 be a hyperplane. Then
the halfspace U0+ contains a hyperplane U1 . Applying leaflessness to U1 yields
95
a hyperplane U2 that is separated from U0 by U1 . Continuing in this manner
gives an infinite nested set in which U0 is initial. The preceding argument
extends this to the desired UBS.
Let U be a UBS. Then there is a partial order ≺ such that for all
U1 , U2 , U3 ∈ U such that U2 separates U1 , U3 , we have U1 ≺ U2 ≺ U3 or
U3 ≺ U2 ≺ U1 , and any two non-crossing hyperplanes in U are ≺-comparable.
∩
Indeed, let f be a consistent section of π such that U ∈U ̸= ∅, and let U1 ≺ U2
if and only if f (U2 ) ⊂ f (U1 ).
Proposition 5.1.4 enables the following proposition about the structure of
unidirectional boundary sets, which in turn allows us to define the simplicial
∪
boundary. If U1 , U2 , . . . , Uk are unidirectional boundary sets, then i≤k Ui
generates an iterated weak nearjoin in ∆X if for all i < j, each element of Uj
crosses all but finitely many elements of Ui .
Proposition 5.1.5 (Structure of boundary sets). Let v be an almostequivalence class of UBSs. Then there exists V ∈ v for which we have
V=
⊔
Uk ,
k∈K
where each Ui is a minimal UBS, and the subgraph of ∆X generated by V is
an iterated (weak) nearjoin of the subgraphs generated by the minimal UBSs
Uk . Hence |K| < ∞.
Moreover, this decomposition is unique up to almost-equivalence: if V ′
⊔
is almost-equivalent to V, and V ′ = k′ ∈K ′ Uk′ ′ is a decomposition of V ′ as a
union of minimal UBSs, then, up to reordering, there is a bijection K → K ′
given by k 7→ k ′ , and Uk is almost-equivalent to Uk′ ′ .
Proof. If V is a minimal UBS, then by definition any UBS U ⊂ V is almostequivalent to V, and the desired decomposition into minimal UBSs exists.
We now proceed inductively. Let U1 be a minimal UBS contained in V,
guaranteed to exist by Proposition 5.1.4 since V is infinite and inseparable.
96
Let V1 = V − U1 . If V1 is finite, then V is almost-equivalent to U1 , and V1
is already minimal.
Let V ∈ V1 be a hyperplane. If V crosses infinitely many hyperplanes
of U1 , then V crosses all but finitely many hyperplanes of U1 since, for all
U ∈ U1 , all but finitely many hyperplanes of U1 lie in a common halfspace
associated to U. Let V1′ ⊂ V1 be the set of hyperplanes in V − U1 that cross
only finitely many elements of U1 . If V1′ is infinite, then since X is locally
finite, V1′ contains an infinite nested set {Vi }i≥0 with initial hyperplane V0 such
that for all but finitely many U ∈ U1 , V0 separates U from each Vi with i ≥ 1.
This implies that U1 ∪ V1′ contains a nested set with no initial hyperplane,
contradicting the fact that V is a UBS. Thus V1′ is finite and can be assumed
to be empty without affecting the almost-equivalence class of V. Hence each
V ∈ V1 crosses all but finitely many elements of U1 , and thus V1 contains an
infinite inseparable set. V1 thus contains a minimal unidirectional boundary
set U2 , by Proposition 5.1.4. Moreover, each U ∈ U2 crosses all but finitely
many of the hyperplanes in U1 .
By orienting each nested subset toward the corresponding point on the
Roller boundary, V can be endowed with a partial order ≺ such that any
two non-crossing hyperplanes are comparable, and if V2 ∈ V separates the
hyperplanes V1 , V3 ∈ V, then V1 ≺ V2 ≺ V3 or V3 ≺ V2 ≺ V1 . This restricts to
a partial ordering on U1 and on U2 , and by the crossing property established
above, U1 ∪ U2 generates a weak nearjoin in ∆X with respect to ≺ .
By induction, we obtain a maximal index set K ⊆ N and a subset
⊔
Uk ⊆ V,
k∈K
where each Uk is a minimal UBS and for all k < n, each U ∈ Un crosses
all but finitely many elements of Uk . For each n ≤ |K|, we can thus find
U1 ∈ U1 , U2 ∈ U2 , . . . , Un ∈ Un such that Ui ⊥Uj for all i ̸= j, and hence
|K| < ∞, by local finiteness of X.
97
Consider the set F = V −
⊔
k∈K
Uk ⊆ V. If F is infinite, then by
Proposition 5.1.4, F contains a UBS, contradicting maximality of K. Hence
F is finite, and we may remove F from V without affecting the almostequivalence class v of V.
Finally, suppose V ′ =
∪
k′ ∈K ′
Uk′ ′ is another such decomposition into
minimal inseparable sets, with V ′ almost-equivalent to V. Suppose there exist
k1′ and k2′ ∈ K ′ and k ∈ K such that Uk has infinite intersection with Uk′ ′
i
for i = 1, 2. Then by inseparability, for i ∈ {1, 2}, we have that Uk ∩
Uk′ ′
i
contains all but finitely many elements of Uk and Uk′ ′ . Without affecting the
i
almost-equivalence class, we can remove finitely many hyperplanes from the
symmetric difference of these sets, and conclude that Uk = Uk′ ′ = Uk′ ′ , and
1
2
hence that k1 = k2 . Thus Uk is almost-equivalent to at most one of the sets
Uk′ ′ .
Note that since each Uk′ ′ is minimal, there is a natural ordering on K ′
such that for k1′ < k2′ , every element of Uk′ ′ crosses all but finitely many
2
elements of Uk′ ′ , and hence |K ′ | ≤ |K|. Suppose Uk is not almost-equivalent to
1
Uk′ ′
′
for any k ∈ K ′ . Then Uk has finite intersection with each Uk′ ′ . But Uk is
∪
infinite and contained in the finite union k′ ∈K ′ Uk′ ′ , a contradiction. Thus Uk
is almost-equivalent to at least one, and thus exactly one, UBS Uk′ ′ .
In the preceding paragraph, we appealed to the following fact: if U1 , U2
are minimal unidirectional boundary sets, then U1 ∩ U2 is infinite if and
only if U1 and U2 are almost-equivalent. Indeed, if U1 is almost-equivalent
to U2 , then, since each is infinite and their symmetric difference finite, their
intersection must be infinite. Conversely, if U3 = U1 ∩ U2 is infinite, then it
contains a UBS U3′ , by Proposition 5.1.4. But U3′ is almost-equivalent to U1 ,
since the latter is minimal. Likewise, U3′ is almost-equivalent to U2 , whence
U1 △U2 is finite.
5.1.2
Simplices at infinity
Definition 5.1.6 (The simplicial complex ∂△ X). The simplicial boundary of
the strongly locally finite CAT(0) cube complex X is the simplicial complex
98
∂△ X constructed as follows. The set of 0-simplices is the set of almostequivalence classes v of minimal UBSs. The n distinct almost-equivalence
classes v1 , . . . , vn of UBSs span an n-simplex [v1 , v2 , . . . , vn ] if and only if there
∪
exist representatives Vi ∈ vi such that i Vi is a UBS. By Proposition 5.1.5,
the simplices of ∂△ X are well-defined and correspond to almost-equivalence
classes of UBSs. See Figure 5–5.
Figure 5–5: A typical UBS corresponding to a 1-simplex at infinity.
For example, the simplicial boundary of R2 is homeomorphic to S 1 .
Indeed, there are four 0-simplices, corresponding to the “positive” and to the
“negative” families of horizontal and vertical hyperplanes. The barycenters
of the four 1-simplices correspond to the sets of hyperplanes crossing the
four geodesic rays emanating from the origin with slope ±1. More generally,
the simplicial boundary of Rn is homeomorphic to S n−1 (it is actually a
hyperoctahedron, by Proposition 5.1.10). The simplicial boundary of a tree is
a totally disconnected set of points corresponding to the topological ends of
the tree.
The simplicial boundary of a CAT(0) cube complex respects convex
subcomplexes in much the same way as the visual boundary of a CAT(0)
space respects convex subspaces [BH99]:
Theorem 5.1.7. Let Y ⊆ X be a convex subcomplex. Then there is a
simplicial embedding ∂△ Y → ∂△ X.
99
Proof. Let W(Y ) be the set of hyperplanes crossing Y , so that each hyperplane of Y is of the form H ∩ Y , where H ∈ W(Y ), and for all H, H ′ ∈ W(Y ),
we have that H ⌣
⊥ H if and only if N (H) ∩ N (H ′ ) ∩ Y ̸= ∅, since Y is convex.
Thus the hyperplanes H ∩ Y and H ′ ∩ Y are separated by H ′′ ∩ Y if and only if
H and H ′ are separated by H ′′ . Thus, if U is a UBS of hyperplanes of Y , the
map W(Y ) ,→ W given by H ∩Y 7→ H shows that U ′ = {H ∈ W : H ∩Y ∈ U}
is a UBS in X.
Let u be the simplex of ∂△ Y represented by U and let u′ be the simplex
of ∂△ X represented by U ′ . Then dim u = dim u′ , by Proposition 5.1.5, since
hyperplanes in U ′ cross if and only if their restrictions to Y cross.
The assignment u 7→ u′ thus determines a simplicial map ∂△ Y → ∂△ X.
Indeed, if u1 , u2 are adjacent simplices in ∂△ Y , then their intersection u3 is a
simplex represented by a boundary set U3 = U1 ∩ U2 , and hence u′3 = u′1 ∩ u′2 .
On the other hand, if u′ is a simplex of ∂△ X and u1 and u2 are simplices
of ∂△ Y mapping to u′ , then u1 and u2 are represented by boundary sets
in Y which extend to almost-equivalent boundary sets in X and are thus
themselves almost-equivalent. Hence the map ∂△ Y → X is injective on the set
of simplices. Thus ∂△ Y simplicially embeds in ∂△ X.
We conclude this section with a few basic properties of ∂△ X, notably
that the presence of a positive-dimensional simplex at infinity precludes
δ-hyperbolicity.
Theorem 5.1.8. The simplicial boundary of X has the following properties:
1. ∂△ X is a flag complex.
2. ∂△ X has dimension d ≤ D − 1, if D = dim X. More generally, each
maximal simplex of ∂△ X is finite-dimensional.
3. ∂△ X is totally disconnected and 0-dimensional if X is δ-hyperbolic.
4. If X = X1 ∪K X2 , where K is a compact subcomplex and X1 , X2 are
nonempty, then ∂△ X = ∂△ X1 ⊔ ∂△ X2 .
100
Proof. Flag complex: Let V0 , . . . , Vn be pairwise-inequivalent UBSs repre∪
senting pairwise-adjacent 0-simplices vi ∈ ∂△ X. Let V = i Vi . If n = 1, then
∪
there is nothing to prove. Suppose by induction that U = i≤n−1 Vi is a UBS.
By Proposition 5.1.5, for all V ∈ Vn , we have that V crosses all but finitely
many elements of U. Hence U ∪ Vn is a UBS.
Finite dimension of simplices: By Proposition 5.1.5, an (n − 1)simplex v = (v1 , . . . , vn ) is represented by a UBS V that generates a subgraph
N of ∆X such that N is a weak nearjoin of n subgraphs generated by minimal
UBSs. Hence ∆X contains an n-clique, and thus n ≤ D. In particular, if ∂△ X
contains an infinite simplex v, then there is an infinite iterated nearjoin, and
hence an infinite clique, in ∆X, contradicting strong local finiteness.
Hyperbolicity: A weak nearjoin N(N, N) in ∆X implies that X contains
arbitrarily large planar grids, since N(N, N) does not have thin bicliques.
Since a 1-simplex in ∂△ X corresponds to such a weak nearjoin, if X is
hyperbolic, then, since ∆X has thin bicliques, ∂△ X cannot contain a 1simplex.
The compactly decomposable case: This is immediate from the
definition: no UBS can consist of hyperplanes crossing K, since K is crossed
by finitely many hyperplanes.
Note that ∂△ X is not in general locally finite; let Fe be a nondiagonal
half-flat, and form X by attaching a non-diagonal quarter-flat Qi to each
vertical geodesic ray of Fe along one of the boundary rays of Qi . Then ∂△ X
has a 0-simplex corresponding to the horizontal hyperplanes of Fe which
is adjacent to each of the infinitely many 0-simplices corresponding to the
vertical hyperplanes of the subcomplexes Qi .
As noted before, if γ : R+ → X is a combinatorial geodesic ray, then the
set W(γ) containing exactly those hyperplanes dual to 1-cubes of γ is a UBS.
It is natural to ask for conditions on X under which the converse holds.
Definition 5.1.9 (Visible simplex at infinity). Let v be an almost-equivalence
class of UBSs. Then v (and the corresponding simplex in ∂△ X) is visible
101
if there exists a geodesic ray γ : R+ → X such that W(γ) ∈ v. The
CAT(0) cube complex X is fully visible if each simplex v of ∂△ X is visible. If
W(γ) ∈ v, we say that the combinatorial geodesic ray γ represents the simplex
v ⊂ ∂△ X.
In Section 5.2, we shall examine what has to happen for a simplex to fail
to be visible. First, we study cubical products and their simplicial boundaries,
using full visibility.
5.1.3
Products and joins
Later, we will see a sense in which boundedness of ΓX corresponds to
a “product-like” structure of X. In [BC11], Behrstock and Charney discuss
the notion of strongly separated walls: the hyperplanes H and H ′ are strongly
separated if they do not contact and, moreover, no hyperplane V contacts
both H and H ′ , i.e. H and H ′ are strongly separated when dΓX (H, H ′ ) > 2.
Under our hypotheses, in the absence of a pair of strongly-separated walls, we
shall see that diam(ΓX) = diam((∂△ X)1 ) = 2. One way in which the graph
(∂△ X)1 can have diameter 2 is to decompose as the join of two nonempty
subcomplexes. We now relate the existence of such a join decomposition to
the existence of a cubical product decomposition of X.
Proposition 5.1.10 is a cubical analogue of Theorem II.9.24 in [BH99],
which characterizes product decompositions of CAT(0) spaces in terms of
spherical join decompositions of their Tits boundaries, and generalizes a result
of Schroeder about Hadamard manifolds ([BGS85], Appendix 4.).
Proposition 5.1.10. Let X be strongly locally finite, fully visible and leafless.
Then the following are equivalent:
1. There exist leafless, fully visible CAT(0) cube complexes X1 and X2 and
a cubical isomorphism
X → X1 × X2 .
2. There exist nonempty disjoint subcomplexes A1 , A2 ⊂ ∂△ X such that
∂△ X ∼
= A1 ⋆ A2 .
102
Proof. (1) ⇒ (2): Let Wi be the set of hyperplanes of Xi . Then W =
W1′ ⊔ W2′ , where
W1′ = {V × X2 : V ∈ W1 }
and, similarly,
W2′ = {X1 × H : H ∈ W2 },
and for all V ∈ W1′ and H ∈ W2′ , the hyperplanes V and H cross. This implies
that any 0-simplex of ∂△ X lies in the image of ∂△ Xi for some i ∈ {1, 2}. From
the above description of the crossings, the claimed join structure is immediate.
Leaflessness of each factor ensures that the Ai are nonempty.
(2) ⇒ (1): This implication is rather more complicated, and we prove it
by “cubifying” the proof of Proposition II.9.25 in [BH99]. Fix a base 0-cube
x ∈ X. For each 0-simplex u ∈ ∂△ X, choose a combinatorial geodesic ray
) = x and W(γ) represents the 0-simplex u. Here
γ : R+ → X such that γ( −1
2
we have appealed to full visibility to choose γ.
For i ∈ {1, 2}, let Ti be the graph in X1 obtained by taking the union
of all rays γ such that γ(− 12 ) = x and γ represents a 0-simplex of A1 . Let
Xi (x) = Xi be the convex hull of Ti , i.e. the smallest subcomplex of X such
that Ti ⊆ Xi (x) and Xi (x)1 ⊆ X1 is convex.
Xi (x) is independent of basepoint in X1 (x): Let x′ ∈ X1 (x) be
a 0-cube, and construct the complex Xi (x′ ) from rays γ ′ based at x′ and
representing 0-simplices of Ai , exactly as was done for Xi . Since X1 (x) is
convex, any hyperplane V that separates x from x′ crosses X1 (x) and X1 (x′ ).
Indeed, if P is a geodesic segment joining x to x′ , then P ⊂ X1 (x) ∩ X1 (x′ ),
since x, x′ ∈ X1 (x) ∩ X1 (x′ ).
If V is a hyperplane crossing X1 (x), then V crosses a geodesic ray γ
emanating from x and representing a 0-simplex of A1 . There is a geodesic
ray γ ′ emanating from x′ that is almost-equivalent to γ. If V crosses γ and
γ ′ , then V crosses X1 (x) ∩ X1 (x′ ). If V separates x from x′ , then V crosses
X1 (x) ∩ X1 (x′ ), as shown above. If not, then V separates γ ′ from an infinite
103
subray α of γ. Hence V crosses all but finitely many of the hyperplanes
that cross γ and γ ′ . As in the proof of Lemma 7.2.2 below, using Sageev’s
construction, we can choose a ray γ1 such that γ1 (− 21 ) = x′ and γ1 is
almost-equivalent to γ, and γ1 contains a 1-cube dual to V . Hence V crosses
X1 (x). Thus X1 (x) and X1 (x′ ) are convex subcomplexes of X such that
W(X1 (x)) = W(X1 (x′ )), and thus X1 (x) = X1 (x′ ).
Verification that X1 ∩ X2 = {x}: Suppose that infinitely many
hyperplanes cross X1 ∩ X2 . By local finiteness and leaflessness, there is a
minimal UBS V such that for all but finitely many V ∈ V, there is a 1cube of X1 ∩ X2 dual to V . Hence ∂△ X contains a 0-simplex in A1 ∩ A2 , a
contradiction. Thus X1 ∩ X2 is compact.
Let V be one of the finitely many hyperplanes that cross X1 ∩ X2 . Then
X1 −X2 and X2 −X1 lie in distinct halfspaces associated to V . By leaflessness,
there exist minimal UBSs V1 , V2 such that Vi consists of hyperplanes that
cross Xi only, and V separates V1 from V2 . Let v1 , v2 be the 0-simplices of
A1 , A2 , respectively, represented by V1 and V2 . Since V1 and V2 are separated
by V ,
d∂△ X (v1 , v2 ) > 1.
Indeed, if v1 , v2 are adjacent, then they are represented by UBS that generate
a weak nearjoin in the crossing graph. On the other hand, since v1 ∈ A1 and
v2 ∈ A2 and ∂△ X ∼
= A1 ⋆ A2 , the simplices v1 and v2 are adjacent, i.e.
d∂△ X (v1 , v2 ) = 1,
a contradiction. Hence no hyperplane crosses X1 ∩ X2 , which therefore consists
entirely of the 0-cube x.
Defining orthogonal projections: Let C ⊆ X be a convex subcomplex. Since X1 is a median graph, and convex subsets of median graphs are
gated [Che00], for each 0-cube y ∈ X, there exists a unique 0-cube ρ(y) ∈ C –
the gate in C of y – such that dX (ρ(y), y) is minimal among all 0-cubes of C.
104
(0)
Hence define orthogonal projections ρi : X(0) → Xi
by letting ρi (y) be the
gate of y in Xi . In particular, ρi is the identity on the 0-skeleton of Xi .
Note that if p, q ∈ X are adjacent 0-cubes, then one of the following
situations occurs. If p, q ∈ Xi , then ρi (p) = p, ρi (q) = q and in particular
the projections are adjacent. If p ∈ Xi and q ̸∈ Xi , then ρi (p) = ρi (q) = q
since q is at distance at least 1 from Xi and at distance exactly 1 from p. If
p, q ̸∈ Xi , then either p is closer to Xi than is q, in which case ρi (p) = ρi (q), or
they are at equal distance from Xi , in which case p, q are separated by a single
hyperplane V that does not separate either from Xi , whence V crosses Xi and
is thus the unique hyperplane separating ρi (p) from ρi (q). Hence we obtain
a map ρi : X1 → X1i that sends 0-cubes to 0-cubes and sends each 1-cube c
isometrically to a 1-cube ρi (c) or collapses c to a 0-cube.
Verification that X2 (x) = ρ−1
1 (x): Let p ∈ X2 and let m be the unique
median of x, p and ρ1 (p). Then m lies on a geodesic joining x to ρ1 (p), so that,
by convexity of X1 , we have m ∈ X1 . On the other hand, m lies on a geodesic
joining x to p so that, by convexity of X2 , we have m ∈ X2 . Therefore,
m ∈ X1 ∩ X2 , whence m = x. Thus x ∈ X1 lies on a geodesic joining p to
ρ1 (p). Since ρ1 (p) is the closest 0-cube of X1 to p, we have ρ1 (p) = x, so that
−1
X2 ⊆ ρ−1
1 (x). On the other hand, since ρ1 is the identity on X1 , if p ∈ ρ1 (x),
then p = x or p ∈ X − X1 . In the latter case, let V be a hyperplane separating
p from X1 . Then by leaflessness of X, there exists a ray γ, containing a 1-cube
dual to V , such that γ(0) = x, γ(t) = p for some t > 0, and γ represents a
simplex u ∈ A2 , from which it follows that x ∈ X2 .
(0)
(0)
Conclusion: Define a map j : X(0) → X1 × X2 by
j(p) = (ρ1 (p), ρ2 (p)),
where p ∈ X is a 0-cube. For all 0-cubes p, q ∈ X, we have
dX1 ×X2 = dX1 (ρ1 (p), ρ1 (q)) + dX2 (ρ2 (p), ρ2 (q));
105
the first term on the right counts the hyperplanes in X that cross X1 and
separate p, q, while the second term counts the hyperplanes that cross X2 and
separate p, q. Since each hyperplane in X crosses exactly one of the convex
subcomplexes Xi , we see that j is an isometric embedding on 0-cubes and
thus extends to an isometric embedding j : X1 → X11 × X12 . Since a CAT(0)
cube complex is determined (up to cubical isomorphism) by its 1-skeleton, it
suffices to show that j is surjective on 0-cubes.
Let (p, q) ∈ X1 × X2 be a 0-cube. Let V1 (p) be the set of hyperplanes
in X separating p from x and let V2 (q) be the set of hyperplanes separating
q from x. Suppose that some V ∈ V2 (q) fails to cross some H ∈ V1 (p). By
leaflessness, there are UBSs V ⊂ V2 and H ⊂ V1 that are separated by V
and H and thus represent simplices v ∈ A2 , h ∈ A1 that are non-adjacent in
∂△ X, a contradiction. Thus each hyperplane in V1 (p) crosses each hyperplane
in V2 (q), and hence there is a 0-cube p′ ∈ X such that the set of hyperplanes
separating p′ from x is precisely V1 (p) ∪ V2 (q). But then ρ1 (p′ ) = p and
ρ2 (p′ ) = q, so that j is surjective.
Remark 5.1.11. To show that (2) ⇒ (1), we require full visibility. For
example, if X is an eighth-flat, then X admits no cubical product structure
since its crossing graph has diameter at least 3. On the other hand, ∂△ X is
the join of two 0-simplices.
5.2
Visible simplices and pairs of simplices
The following lemma is a simple application of Sageev’s construction.
Lemma 5.2.1. Let U be a nonempty, finite, inseparable set of hyperplanes
containing no facing triple. Then there exists a geodesic segment P → X such
that W(P ) = U.
Lemma 5.2.1 is used in the proof of Theorem 5.2.2 to build a sequence of
geodesic segments from which we then attempt to construct a geodesic ray as
a limit.
106
5.2.1
Maximal simplices are visible
The main theorem of this section is the following:
Theorem 5.2.2 (Recognition of geodesic rays). Let X be a strongly locally
finite CAT(0) cube complex, and let V ⊆ W be a UBS of hyperplanes. Then
there exists a geodesic ray γ : R+ → X such that W(γ) ⊇ V, i.e. V extends to
a visible UBS. In particular, every maximal simplex v of ∂△ X is visible.
Proof. We attempt to construct the ray γ in a natural way and, if we fail,
construct a simplex u of ∂△ X that properly contains v. Define a section
f : V → V + of π|V and a partial ordering ≺ on V as in Lemma 5.1.2.
A sequence of segments: By definition, there exists at least one ≺minimum V0 ∈ V. Fix V0 and choose V0 ⊂ V1 ⊂ . . . ⊂ Vr ⊂ . . . such that
∪
|Vr | = r + 1, each Vr is inseparable, and r≥0 Vr = V. Also, make this choice
so that the hyperplane V0 ∈ V0 is ≺-minimal in V0 and has the property that
∪
every element of r Vr either lies in f (V0 ) or crosses V0 . Note that if V0 is
maximal and minimal, then every hyperplane crosses V0 , and we can choose
V0 = {V0 }.
By Lemma 5.2.1, for each r ≥ 0, there exists a 0-cube cr ∈ N (V0 ) and a
geodesic segment Pr → X emanating from cr such that W(Pr ) = Vr .
∪
Let Ω be the graph whose vertex-set is r≥0 Rr , where Rr is the set of
geodesic segments Pr whose 1-cubes are dual to exactly the hyperplanes in
Vr . Let an edge join Pr ∈ Rr to Pr+1 ∈ Rr+1 if and only if Pr ⊂ Pr+1 . The
graph Ω is infinite and locally finite. Indeed, local finiteness follows from local
finiteness of X. If Pr is a geodesic segment crossing Vr and extending to Pr+1 ,
then let c′ be the 1-cube of Pr+1 − Pr . If V is dual to the terminal 1-cube of
Pr adjacent to c′ , and V1 , . . . , Vn ∈ Vr+1 − Vr are dual to 1-cubes c1 , . . . , cn
extending Pr , then since V contains no facing triple, the set V, V1 , . . . , Vn
contains a collection of n pairwise-crossing hyperplanes, and thus n is bounded
above by the dimension of the maximal cube containing c′ , which is finite by
107
local finiteness of X.1 Hence each Pr extends in finitely many ways in each
direction, and thus Ω is an infinite, locally finite graph.
Taking a limit to obtain a ray: If Ω has an infinite connected
component, then by König’s lemma [Kön36], there is an infinite path γ → X
∪
that crosses every hyperplane of r≥0 Vr = V ′ exactly once. Hence γ is
a geodesic ray whose initial 1-cube is dual to the given hyperplane V0 .
Moreover, every hyperplane lying in f (V0 ) is dual to a 1-cube of γ. Hence f is
visible.
When the limit does not exist: If every component of Ω is finite, then
for each n and each ϕ ∈ Rn , there exists a maximal k(ϕ) ≥ n such that ϕ
extends to an element of Rk(ϕ) . Suppose there is a hyperplane U (ϕ) separating
∪
ϕ from infinitely many ψ ∈ m>k(ϕ) Rm . Since it separates the initial point of
ϕ from the initial point of infinitely many such segments ψ, and ϕ and each
such ψ emanate from N (V0 ), the hyperplane U (ϕ) crosses V0 . Moreover, since
∪
U (ϕ) separates ϕ from each ψ ∈ m≥k(ϕ) Rm , the hyperplane U (ϕ) ̸∈ V, since
each hyperplane of V is dual to a 1-cube of infinitely many of the segments ψ.
Hence we can choose a subsequence (ij )j≥1 and, for each j ≥ 1, a segment
ϕj ∈ Rij and a hyperplane Uj ̸∈ V ′ that separates ϕj from ϕk for all k > j,
and thus crosses V0 . See the left side of Figure 5–6. Note that if V ∈ Vij , then
V crosses ϕk for all k ≥ j, and hence V crosses Uk for all k ≥ j. Thus V ′ and
{Uj }j≥0 yield a weak nearjoin in the crossing-graph ∆X, where the partial
∪
ordering on V ′ comes from the grading V ′ = r Vr and the partial ordering on
the set of Uj is the obvious one.
On the other hand, if some hyperplane U separates Ui from Uj with i < j,
then U must cross every hyperplane crossed by both Ui and Uj , and hence
1
Note that the above argument shows that Ω is uniformly locally finite
when X is finite-dimensional. To prove the claim as stated, we do not need
the pairwise-crossing conclusion above; we need only observe that there are
finitely many cubes containing c′ .
108
Figure 5–6: The configuration of hyperplanes and segments arising when there
is no dual ray associated to a simplex at infinity.
adding U to the set {Uj }j≥0 does not affect its status as a weak nearjoin.
Hence there is an infinite, inseparable, partially-ordered set U, containing all
of the Uj , such that the crossing graph ∆X of X contains a weak nearjoin Λ of
V and U.
Finding a larger simplex: The set V ′ = V ∪ U is a UBS containing
V and representing a simplex v ′ of ∂△ X such that v ′ properly contains the
simplex v represented by V. In particular, v is not maximal. Hence each
maximal simplex of X is visible. Moreover, by Theorem 5.1.8 and local
finiteness of X, the simplex v is contained in a maximal simplex and hence V
extends to a UBS almost-equivalent to W(γ) for some geodesic ray γ.
Proof that U (ϕ) exists: Suppose ϕ ∈ Rn does not extend to any
∪
ψ ∈ Rm for m > k(n). Either infinitely many ψ ∈ m>k(n) Rm are separated
from ϕ by a hyperplane U ̸∈ V, or we are in the following situation for all but
finitely many such ψ. Let m > k(n) ≥ n + 1, so that ψ = ϕ′ α, where ϕ′ is
a length-n geodesic segment crossing Vn and α is a geodesic segment crossing
Vm − Vn . Then ϕ(n) and ϕ′ (n) are separated by a hyperplane U that crosses
each of the hyperplanes in Vm − Vn , for otherwise ϕ would extend to a segment
ϕα ∈ Rm . If U ̸∈ V, then we are done, since U must then separate ϕ from ψ.
Otherwise, U crosses ϕ or ϕ′ and thus U ∈ Vn . Hence ϕ ∩ ϕ′ ̸= ∅, so that some
sub-segment of ϕ extends to a segment in Rm . If this holds for arbitrarily
large values of m, with ϕ fixed, then Lemma 2.1.1 yields a ray whose initial
109
0-cube lies on ϕ and whose set of dual hyperplanes is V. Thus we may suppose
that there exists U ̸∈ V that separates ϕ and ψ.
It follows from Theorem 5.2.2 that if V is a visible UBS, then any UBS V ′
containing V is also visible, since V ′ represents a simplex at infinity containing
the simplex represented by V.
Proposition 5.2.3. For each simplex v = [v0 , v1 , . . . , vn ] of ∂△ X such that vi
is visible for all 0 ≤ i ≤ n, there exists a combinatorial isometric embedding
Y → X with Y ∼
= R+ n+1 such that the image of the induced embedding
∂△ Y → ∂△ X is the n-simplex v.
Proof. For 0 ≤ i ≤ n, let Vi be a UBS representing vi . Since vi ̸= vj for
i ̸= j, we can choose the representatives Vi so that Vi ∩ Vj = ∅ whenever
i ̸= j. Since each vi is a 0-simplex, we may assume, by modifying Vi within its
almost-equivalence class, that no V ∈ Vi crosses all but finitely many V ′ ∈ Vi .
Exhibiting a join in ΓX: By definition, for any pair i ̸= j, the sets Vi
and Vj generate a subgraph Nij ⊆ ∆X containing a weak nearjoin of Vi and
Vj : each Vj ∈ Vj crosses all but finitely many of the hyperplanes in Vi . Let
γi → X and γj → X be geodesic rays dual to Vi and Vj respectively, whose
existence is guaranteed by full visibility. We can assume that the hyperplanes
Vi0 , Vj0 , dual to the initial 1-cubes of γi and γj respectively, actually cross, by
removing finitely many hyperplanes, if necessary, from Vi and appealing to the
fact that Vi ⊔ Vj generates (at least) a weak nearjoin in ∆X. Hence we can
choose γi and γj to have the same initial 0-cube. Moreover, since their sets
of dual hyperplanes are disjoint, γi and γj intersect in a single 0-cube xo and
their union is isometrically embedded.
By local finiteness of X, for all n ≥ 0, there exist hyperplanes Vin ∈
Vi , Vjn ∈ Vj such that at least n elements of Vi separate xo from Vin , and at
least n elements of Vj separate xo from Vjn . Moreover, these hyperplanes can
be chosen so that Vin and Vjn cross. Now let Ani → N (Vin ), Anj → N (Vjn ) be
geodesic segments with a common endpoint in N (Vin ) ∩ N (Vjn ) and let Bin , Bjn
110
be initial segments of γi and γj respectively, chosen so that Bin Ani Anj Bnj is a
closed embedded path in X bounding a disc diagram Dn → X that is minimal
for all choices of Ani , Anj , Bin , Bjn .
No dual curve in Dn travels from Ani to Anj by minimality of area. Indeed,
if K were such a dual curve, we could perform hexagon moves and removals of
spurs to modify Ani and Anj , without affecting the hyperplane-carriers to which
they map, yielding a new diagram with either lower area or shorter boundary
path. (This procedure of fixed carrier diagrams is explained in more detail
in [Hag11].) Moreover, no dual curve travels from Bni to Bjn , since Vi ∩ Vj = ∅.
Finally, if K, K ′ are dual curves mapping to the same hyperplane, then K, K ′
must both cross Bin , or they must both cross Bjn , and this is impossible since
those paths are geodesic. By construction, at least n dual curves travel from
Ani to Bjn and at least n dual curves travel from Anj to Bin , and each dual
curve of the first of these types crosses each dual curve of the second of these
types. Moreover, each Vk ∈ Vi and each Vm ∈ Vj appears as such a dual
curve in Dn for all sufficiently large values of n, since Vk crosses only finitely
many elements of Vi . Thus Vk crosses Vm for all Vk ∈ Vi and all Vm ∈ Vj . By
∏
Proposition 2.3.8, there is an isometric embedding ni=0 γi → X.
5.2.2
Cube complexes are “optical spaces”
Pairs of simplices in ∂△ X are often represented by bi-infinite geodesics,
in the sense that, given two simplices v, v ′ , there exists a bi-infinite geodesic
α that is the union of two geodesic rays, one of which represents each of the
simplices v and v ′ . This fact will be important in Chapter 7, and the following
theorem illustrates the two techniques – folding and truncation – used to find
such bi-infinite geodesics.
Definition 5.2.4 (Visible pair, optical space). Let u, v be distinct simplices of
∂△ X. Suppose that u, v are respectively represented by UBSs U, V such that
U ∪ V = W(γ) for some bi-infinite combinatorial geodesic γ : R → X. Then
u, v is a visible pair. If for any two 0-simplices u, v ∈ ∂△ X with u ̸= v, the pair
u, v is visible, then X is a optical space.
111
Theorem 5.2.5. Let v, v ′ be a pair of distinct visible simplices of ∂△ X. Then
either there exists a combinatorial geodesic α : R → X such that W(α) =
V ∪ V ′ , where V (respectively, V ′ ) is a UBS representing v (respectively, v ′ ), or
v ∩ v ′ ̸= ∅. Hence, if X is fully visible, then X is an optical space.
Proof. Choose geodesic rays γ, γ ′ : R+ → X such that γ(− 21 ) = γ ′ ( 12 ) and
W(γ), W(γ ′ ) respectively represent the simplices v and v ′ . Let U be the set of
hyperplanes U such that U is dual to a 1-cube of γ and to a 1-cube of γ ′ . We
shall show that, if U is finite, then γ ∪ γ ′ can be modified to produce a single
geodesic representing the pair v, v ′ , and if U is infinite, then γ and γ ′ represent
adjacent simplices at infinity. We define two moves, folding and truncation,
that finitely modify the subcomplex γ ∪ γ ′ .
Truncation: Suppose γ ∩ γ ′ contains an embedded finite path P with
|P | ≥ 1. Let γ1 , γ1′ be the shortest subpaths of γ, γ ′ , respectively, containing
P . See the left side of Figure 5–7. Let γ̄ be the closure of γ − γ1 and let γ̄ ′
be the closure of γ ′ − γ1′ . Since γ̄ ∩ γ̄ ′ contains the terminal 0-cube of P , the
subcomplex γ̄ ∪ γ̄ ′ is a connected union of two geodesic rays. Since γ1 , γ1′ each
have finitely many dual hyperplanes, γ̄ ∪ γ̄ ′ represents the pair of simplices
v, v ′ at infinity. Note that the set U of hyperplanes U that cross both γ̄ and γ̄ ′
Figure 5–7: Truncation of the union of two geodesic rays.
is properly contained in U.
Folding: Folding takes as its input the concatenated geodesic rays γ
and γ ′ and returns geodesic rays γ̂ and γ̂ ′ that are dual to the same sets of
hyperplanes as γ and γ ′ , respectively, with the property that γ̂ and γ̂ ′ coincide
in the initial segments preceding the 1-cubes dual to some U ∈ U.
112
Let U ∈ U be a hyperplane that crosses γ and γ ′ . Let A, A′ be subpaths
of γ, γ ′ respectively joining γ(0) = a to 0-cubes b, b′ of N (U ), each of
which is separated from a by U , and let B → N (U ) be a geodesic segment
joining b to b′ . Let m ∈ X(0) be the median of a, b, b′ . Let Â be the unique
combinatorial geodesic segment passing through m and joining a to b, and
define Â′ identically for b′ . Also, let B̂ be the unique geodesic passing through
m and joining b to b′ . Since N (U ) is convex, B̂ → N (U ). See Figure 5–8.
Since b and b′ lie in the same halfspace of U , the path B̂ cannot contain a
Figure 5–8: Folding.
1-cube dual to U . On the other hand, each of the paths Â and Â′ crosses U ,
and hence the unique 1-cube dual to U in the tripod Â ∪ Â′ ∪ B̂ must lie in the
segment P = Â ∩ Â′ .
Let γ̂ be obtained from γ by replacing A by Â and let γ̂ ′ be the geodesic
ray obtained from γ ′ by replacing A′ by Â′ . Then, since P ⊂ γ̂ ′ ∩ γ̂ has
length at least 1, the new pair of rays satisfies all of the desired conditions. In
particular, finitely many folding operations have no effect on the simplices v, v ′
at infinity.
The case |U| < ∞: If U is finite, then for each U ∈ U, we may fold
and then truncate to produce a new pair γ̄, γ̄ ′ with strictly fewer common
hyperplanes than the pair γ, γ ′ . If γ ∩ γ ′ contains a nontrivial subpath P , then
|U| ≥ 0. Hence, by performing finitely many truncations, we may assume that
γ and γ ′ are a pair of geodesic rays representing v and v ′ respectively, such
113
that each hyperplane crosses γ ∪ γ ′ at most once, and γ( −1
) = γ ′ ( −1
). Define
2
2
a map α : R → X by α(t) = γ(−t) for t ≤
−1
2
and α(t) = γ ′ (t) for t ≥
−1
.
2
By
definition, α is a combinatorial geodesic and the bi-directional boundary set
W(α) represents the pair of simplices v, v ′ at infinity.
The case |U| = ∞: If U is almost-equivalent to W(γ), then v ⊆ v ′ by
definition, and hence d∂△ X (v, v ′ ) = 0. Thus we may assume that W(γ) = S ∪U
and W(γ ′ ) = S ′ ∪ U, where S and S ′ are disjoint infinite sets of hyperplanes,
inheriting partial orderings from W(γ) and W(γ ′ ) respectively. U also inherits
a partial ordering. If U1 , U2 ∈ U and some hyperplane W separates U1 from
U2 , then W must cross γ and γ ′ . Hence W ∈ U, and thus U is inseparable.
Thus there is a simplex u of ∂△ X represented by u.
Let S ∈ S, so that S is dual to a 1-cube of γ but is not dual to a 1-cube
of γ ′ . Then S separates an infinite subray γ1 of γ from γ ′ and hence crosses all
but finitely many of the hyperplanes in U, namely those U ∈ U that cross γ1 .
This implies that S and S ′ are inseparable.
Hence S, S ′ are UBSs representing simplices s ⊂ v, s′ ⊂ v ′ at infinity, and
each of s, s′ is adjacent to u. Thus v ∩ v ′ contains u.
We summarize as follows: if v, v ′ are both visible but no combinatorial
geodesic represents the pair v, v ′ , then either all but finitely many hyperplanes
crossing γ cross γ ′ , and v ⊆ v ′ , or u is a common proper face of v and v ′ .
In particular, if v, v ′ are distinct visible 0-simplices, then the pair v, v ′ is
visible.
5.2.3
Rank one notions and fellow-traveling with hyperplanes
There are several different ways to define a rank-one geodesic in a CAT(0)
cube complex, all of which become equivalent in a cocompact situation. We
choose the definition that most obviously agrees with the usual notion when
the geodesic in question is an axis of a rank-one isometry.
Definition 5.2.6 (Rank one geodesic ray, rank-one geodesic). Let γ : R+ →
X be a combinatorial geodesic ray. Then γ bounds a fractional flat if there
exists an isometrically embedded fractional flat F ⊆ X such that γ ⊂ F.
114
Figure 5–9: Each of the two colored rays at left bounds an eighth-flat. At
right is part of a ray in R+ 3 bounding an eighth-flat.
Figure 5–10: The top-left geodesic bounds a diagonal half-flat. The bottom
left pair of rays together bound a diagonal quarter-flat, and each bounds an
eighth-flat. The two colored rays in R+ 3 co-bound a diagonal quarter-flat.
Note that F can be a flat, diagonal or non-diagonal half-flat, diagonal or
non-diagonal quarter-flat, or eighth-flat. Similarly, if γ : R → X is a biinfinite combinatorial geodesic, then γ bounds a fractional flat if there exists
an isometrically embedded fractional flat F such that γ ⊂ F ⊆ X. In this
situation, F can be a non-diagonal quarter-flat, a diagonal or non-diagonal
half-flat, or a flat. Figure 5–9 shows some rays and geodesics bounding
fractional flats. Figure 5–10 shows some other configurations.
If γ is a geodesic ray or bi-infinite geodesic in X that does not bound a
fractional flat, then γ is rank-one.
115
Let v ⊆ ∂△ X be a maximal simplex containing the simplex represented
by the unidirectional boundary set W(γ), where γ is a geodesic ray. We will
see momentarily that, if dim v ≥ 1, then γ bounds a fractional flat. On the
other hand, if v is a isolated 0-simplex, then γ must be rank-one, since the
set of hyperplanes dual to an isometric fractional flat contains at least two
inequivalent minimal unidirectional boundary sets generating a weak nearjoin
in ∆X. In other words:
Lemma 5.2.7. Let X be a strongly locally finite CAT(0) cube complex and let
γ be a geodesic ray in X. Let v be the simplex of X represented by γ. Then γ
is rank-one if and only if v is an isolated 0-simplex.
Proof. Suppose that γ is not rank-one, and let F be a fractional flat containb be the convex hull of F. Then ∂ F
b contains a 1-simplex u ∼
ing γ. Let F
= ∂△ F
△
b ⊆ ∂ X by Theorem 5.1.7, so
such that u contains v. On the other hand, ∂△ F
△
that v is not isolated, since it lies in the image of u.
Conversely, suppose that v is not an isolated 0-simplex. If v is a 0simplex, then v is contained in a maximal simplex u of dimension at least
1. Otherwise, v itself has dimension at least 1. We shall show that, if u =
[u0 , . . . , un ] is a maximal simplex with n ≥ 1, then for every visible face u′ of
u, every ray representing u′ bounds a fractional flat. This implies that γ is not
rank-one whenever v is not an isolated 0-simplex.
If n = 1, let α represent u0 and let U1 be a unidirectional boundary
set such that each U ∈ U1 crosses all but finitely any elements of W(α).
An application of Sageev’s construction, or a disc diagram argument in
conjunction with Lemma 2.1.1, shows that X contains an isometrically
embedded eighth-flat containing α, which is therefore not rank-one. The other
ray bounding this eighth-flat can be taken to be any ray β with W(β) =
U1 ∪ W(α), so that any ray representing u = [u0 , u1 ] is not rank-one. The more
general claim now follows by induction on n.
116
The following may be proven by directly constructing a fractional flat and
a map to X.
Lemma 5.2.8. Let X = Q1 × Q2 , where Qi is an unbounded locally finite
CAT(0) cube complex for i ∈ {1, 2}. If α is a bi-infinite geodesic, then there is
an isometrically embedded non-diagonal fractional flat containing α.
Also, if F is an isometric fractional flat, then diam(ΓF) < ∞. In
particular, if γ bounds a fractional flat, then diamΓX (ProjΓ γ) < ∞. On the
other hand, the converse does not hold if, for example, γ lies in a uniform
neighborhood of a hyperplane. This is discussed in more detail in Chapter 6,
and motivates the projection trichotomy. The other notion needed for the
projection trichotomy is: the geodesic ray or bi-infinite geodesic γ fellowtravels with the hyperplane H if there exists some R < ∞ such that for all
0-cubes x ∈ γ, we have dX (x, N (H)) ≤ R.
Let G act on X. By a lovely result of Haglund in [Hag07], each g ∈ G,
viewed as a cubical automorphism of X, is either elliptic, i.e. gx = x for some
x ∈ X, or hyperbolic, i.e. there exists τg > 0 and a combinatorial geodesic
γ : R → X such that (gγ)(t) = γ(τg + t) for all t ∈ R . The geodesic γ
is a combinatorial axis for g. The hyperbolic isometry g ∈ G is rank-one if
for some – and hence for any – axis γ for g, the geodesic γ does not bound a
half-flat.
5.3
Comparison to other boundaries
Through simple examples, we distinguish ∂△ X from the other boundaries
that X possesses as a cube complex or as a CAT(0) space.
Example 5.3.1 (The visual boundary). The space (X, dX ) is a CAT(0) space,
and thus has a visual boundary ∂∞ X, which is nicely described in e.g. [BH99].
Roughly speaking, the points of ∂∞ X are equivalence classes of geodesic
rays emanating from a fixed basepoint, where two rays are equivalent if they
fellow-travel.
Let X be a regular tiling of H2 by 2-cubes, arising, for example, as
the universal cover of a tiling of a closed hyperbolic surface by 2-cubes.
117
Then X is quasi-isometric to H2 , and since the homeomorphism type of the
visual boundary of a hyperbolic metric space is quasi-isometry invariant,
∂∞ X ∼
= ∂∞ H2 . The visual boundary of H2 can easily be shown to be
homeomorphic to S 1 . On the other hand, ∂△ X is a totally disconnected
simplicial complex, by Theorem 5.1.8, and hence ∂△ X is not in general
homeomorphic to ∂∞ X.
There are also non-hyperbolic examples. A major difficulty with the
visual boundary is that, for CAT(0) spaces that are not hyperbolic, the
homeomorphism type of the visual boundary is not in general quasi-isometry
invariant. The first example of this phenomenon is a 2-dimensional CAT(0)
cube complex X, which is the universal cover of the Salvetti complex of the
right-angled Artin group whose underlying graph is an interval subdivided into
three 1-cells. Croke and Kleiner showed that small perturbations of dX give
new CAT(0) metrics with respect to which the homeomorphism type of ∂∞ X
changes [CK]. On the other hand, ∂△ X does not care about the particular
choice of CAT(0) metric, since ∂△ X is defined entirely combinatorially: the
collection of unidirectional boundary sets and combinatorial geodesic rays
does not change as the CAT(0) metric is perturbed. Hence ∂△ X fails to be
homeomorphic to ∂∞ X for at least one choice of CAT(0) metric on X.
Example 5.3.2 (The Tits boundary). The Tits boundary of (X, dX ) is
the classical boundary that most closely resembles ∂△ X, as we shall see in
Chapter 7. Indeed, when X is fully visible, we shall describe a sense in which
∂△ X is a “combinatorial Tits boundary” for X.
Example 5.3.3 (The Roller boundary). We saw in Section 2.2.4 that the
Roller boundary ∂R X is always compact. On the other hand, if X is a tree
with infinitely many ends, then ∂△ X is an infinite, totally disconnected set
and hence is not homeomorphic to ∂R X.
118
5.4
Problems on the simplicial boundary
Problem 5.4.1 (Quasi-isometry invariance). As alluded to in Example 5.3.1,
∂∞ X is not a quasi-isometry invariant of X. On the other hand, a quasiisometry invariant boundary is an extremely useful tool from a group-theoretic
point of view; a wide variety of results are proved, for example, using the
Gromov boundary of a hyperbolic group G (i.e. the visual boundary of any
space on which G acts properly and cocompactly by isometries). Let X, Y
be CAT(0) cube complexes and let q : X1 → Y1 be a quasi-isometry. What
is the relationship between ∂△ X and ∂△ Y? The best possible answer would
be the existence of a simplicial isomorphism, or at least a homeomorphism
∂△ X → ∂△ Y. This would give each cocompactly cubulated group a welldefined boundary. More conservatively: let G act properly and cocompactly
on the CAT(0) cube complexes X and Y. Is ∂△ X homeomorphic to ∂△ Y?
Remark 5.4.2 (Cubulated Bieberbach groups). A test case for Problem 5.4.1
comes from the crystallographic groups. A countable discrete group G is
crystallographic if there is a proper, cocompact action of G on Rn for some
n < ∞. Generally speaking, it is easy to find a G-invariant system of walls in
Rn satisfying the linear separation property, so that G acts properly on the
dual cube complex. However, it is more difficult to obtain a cocompact action;
the (3, 3, 3) triangle group is already an interesting example, in which n = 2
but the natural dual cube complex is R3 .
The hyperoctahedron On is the triangulation of S n defined by letting O0
be a pair of 0-simplices and letting On = On ⋆ O0 . Note that ∂△ Rn ∼
= On−1
by Proposition 5.1.10. If G acts properly and cocompactly on X, then X
must be quasi-isometric to Rn , and thus quasi-isometric to Rn . In [Hag], it is
shown that, if G is an n-dimensional crystallographic group that acts properly
and cocompactly on a CAT(0) cube complex X, then ∂△ X is necessarily
isomorphic to On−1 . In other words, the crystallographic groups provide a
class of examples for which the second question posed in Propblem 5.4.1 has a
positive answer.
119
CHAPTER 6
Bounded contact graphs and their applications
In this chapter, X is a locally finite, leafless CAT(0) cube complex.
Beginning in the second section, we shall also require strong local finiteness,
in order to use the simplicial boundary. The main theorem is Theorem 6.2.3,
which gives conditions under which ΓX is quasi-isometric to a point, both in
terms of ∂△ X and in terms of the cubical structure of X itself. We use these
ideas to re-interpret the rank-rigidity theorem of Caprace and Sageev, giving a
modified proof.
Definition 6.0.3 (Pseudo-product). X is the pseudo-product of the cube
complexes Q1 and Q2 , denoted by X ∼
= Q1 Q2 , if all of the following hold:
1. There is a convex embedding a : Q1 → X, and the image of Q1 is
crossed by an infinite set V1 of hyperplanes. There is a restriction
quotient q1 : X → Q1 such that a is a section of q1 , and q1 is obtained by
restricting to the set V1 of hyperplanes.
2. There is a restriction quotient q2 : X → Q2 obtained by restricting to the
set V2 = W − V1 of hyperplanes. Moreover, V2 is infinite.
3. There is a cubical isometric embedding e : X → Q1 × Q2 for which the
convex hull of e(X) is all of Q1 × Q2 . Also, if π1 : Q1 × Q2 → Q1 is the
projection, then π1 ◦ e ◦ a is the identity on Q1 .
4. For each H ∈ V2 , there are infinitely many V ∈ V1 such that H⊥V in X.
Note that, if X ∼
= Q1 × Q2 , then X ∼
= Q1 Q2 . However, for example, an
eighth-flat is a pseudo-product of two copies of R+ , but is not a product of
any two cube complexes since its crossing graph has diameter at least 3.
Theorem 6.2.3 relates boundedness of ΓX to boundedness of (∂△ X)1
and to pseudo-product decompositions of X. The first step in proving
120
Theorem 6.2.3 is to study the combinatorial geodesic rays that project to
unbounded rays in the contact graph.
6.1
The projection trichotomy
The main tool used to prove Theorem 6.2.3 is the projection trichotomy,
which takes two forms. Although we shall use the second, the first is of
independent interest. The projection trichotomy also immediately implies
Corollary 6.3.1, which describes the behavior of rank-one isometries as
isometries of the contact graph.
Theorem 6.1.1 (Projection trichotomy). Let X be a locally finite CAT(0)
cube complex, and let γ : R+ → X be a combinatorial geodesic ray. Let ProjΓ γ
be the projection to the contact graph, and let Λγ be the full subgraph of ΓX
generated by ProjΓ γ. Then one of the following holds:
1. The inclusions ProjΓ γ, Λγ ,→ ΓX are quasi-isometric embeddings.
2. For all m ≥ 0, there exists an embedding Km,m → Λγ.
3. There exists R < ∞ such that, for all t ≥ 0, there exists a hyperplane W
such that γ ∩ NR (W ) has length at least t.
Moreover, exactly one of the following holds:
4. γ has unbounded projection.
5. γ bounds a fractional flat, or γ fellow-travels with a hyperplane, or both.
Remark 6.1.2. A heuristic picture of the projection trichotomy appears in
Figure 6–1. In our applications, we shall always use the assertion that (4) and
(5) are mutually exclusive. Note that (4) is a weaker version of (1), while (5)
is a stronger version of the disjunction of (2) and (3).
Proof of Theorem 6.1.1. We prove the first assertion, about statements (1)(3), using disc diagrams. We then establish the assertion about statements
(4)-(5) by a simpler argument.
Fix a parametrization γ : R+ → X, and let Hi be the hyperplane dual to
the 1-cube γ([i − 12 , i + 12 ]), for i ≥ 0, so that
ProjΓ γ = H0 ⌣H
⊥ 1 ⌣H
⊥ 2⌣
⊥ ....
121
Figure 6–1: At left are the situations described in statements (2) and (3) of
Theorem 6.1.1. The assumption that γ projects to a finite-diameter path in Γ
leads to one of the situations (4) or (5) shown at right.
The graph λγ is the full subgraph of ΓX generated by the set {Hi }i≥0 of
vertices of ΓX.
Setting up disc diagrams: For each i ≥ 0, choose a geodesic
ψi = H0 ⌣U
⊥ 1⌣
⊥ . . . ⌣U
⊥ ki −1 ⌣H
⊥ i
of ΓX joining H0 to Hi , where ki = dΓX (H0 , Hi ). For each 0 ≤ j ≤ ki , let
ρj → N (ψ(j)) be a geodesic segment in the carrier of H0 , Hi or Uj , chosen
so that the ρj are concatenable, i.e. there is a path Pi = ρ0 ρ1 . . . ρki with the
same endpoints as Qi = γ([− 12 , i + 21 ]). Let Di → X be a disc diagram bounded
by Pi Q−1
i . The diagram Di is chosen subject to the following minimality
conditions. First, Di has minimal area among all disc diagrams with the given
boundary path. Second, the geodesic segments ρi are chosen so that Pi results
in a lowest-area possible diagram Di , among all paths Pi constructed in the
above manner from the data γ, ψi . Finally, ψi is chosen among all geodesics
of ΓX joining H0 to Hi in such a way that the area of Di is minimized among
all such choices. Figure 6–2 shows Di . These minimality assumptions are very
similar to the minimality conditions on a diagram with fixed carriers, used
in [Hag11], and we use identical reasoning to rule out certain configurations of
dual curves in Di .
Destinations of dual curves: Let K be a dual curve in Di emanating
from Qi . Then K ends on Pi , since Qi is a geodesic segment and thus contains
122
Figure 6–2: The diagram Di in the proof of the first assertion of the projection trichotomy.
at most one 1-cube dual to each hyperplane. This type of impossible dual
curve is shown at the bottom of Figure 6–3.
Let K be a dual curve emanating from ρj . Then K cannot end on ρj ,
since ρj is a geodesic segment. Moreover, as discussed below, K cannot end
on ρj±1 , since that would result in a lower-area choice of Di . Furthermore,
K cannot end on ρj±2 , for otherwise the hyperplane to which K maps could
replace ψi (j ± 1), resulting in a diagram Di′ ( Di fulfilling all of the defining
criteria of Di and thus contradicting the minimal-area choice of ψi . Finally,
if K travels from ρj to ρk with |j − k| > 2, then the hyperplane to which
K maps provides a shorter path than ψi from H0 to H1 . These impossible
configurations are shown in Figure 6–3. Hence every dual curve travels from
Pi to Qi .
If K1 , K2 , emanating from ρj , are distinct dual curves, then they do
not cross. Indeed, should K1 , K2 cross, a sequence of hexagon moves would
allow use to choose a lower-area diagram Di , with a new choice of ρj →
N (ψi (j)); see [Wisa]. If K travels from ρj to ρj±1 , then either some dual curve
emanating from ψi (j) or ψi (j + 1) crosses K, or ψi (j) ∩ ψi (j + 1) contains more
than one 0-cube, and we can fold and remove spurs, as in [Hag11] to produce
123
Figure 6–3: Each dual curve, or configuration of dual curves, is impossible.
The numbered configurations are illustrated in the next two figures; the unnumbered ones contradict minimality of Di or ψi for straightforward reasons.
a new diagram Di′ with lower area or shorter boundary path, without affecting
Qi or ψi . See Figure 6–4.
Figure 6–4: Di does not contain “triangles along the boundary”. The singlearrowed path precludes crossing dual curves emanating from the same “syllable” of Pi , while the double-arrowed path precludes dual curves starting and
ending on consecutive syllables.
We conclude that the dual curves in Di are as in Figure 6–5: each travels
from Pi to Qi and thus maps to a hyperplane crossing γ. In particular, any
two distinct dual curves map to distinct hyperplanes. Moreover, dual curves
emanating from ρj do not cross in Di . Also, note that since ψi is a geodesic of
ΓX, if Kj is a dual curve emanating from ρj and Km emanates from ρm , and
Kj contacts Km in Di , then |j − m| ≤ 2. Indeed, if |j − m| > 3, then there is
a path shorter than ψi from H0 to Hi . If |j − m| = 3, then there is a different
choice of geodesic in ΓX, using the hyperplanes to which Kj , Km map, that
results in a lower-area choice of Di . Finally, |ρj | ≥ 1 for each j, for otherwise
124
Figure 6–5: The dual curves in Di .
we have Uj−1 ⌣U
⊥ j+1 , contradicting minimality of |ψi |.
Plan for using Di : Note that since all dual curves travel from Pi to Qi ,
ki
∑
|ρj | = |Qi | = | ProjΓ Qi | + 1.
j=0
Let λi = max0≤j≤ki |ρi |, which is at least 1, as was observed above. Let
λ = supi λi . We shall show that, if λ < ∞, then the projection of γ to ΓX
is a quasi-isometric embedding. If λ = ∞, then we shall use the diagrams Di
to exhibit either arbitrarily large bicliques in Λγ or a constant R such that γ
contains arbitrarily long segments that R-fellow travel with hyperplanes.
A quasi-isometric embedding when λ < ∞: Suppose λ < ∞. Then
we have
| ProjΓ Qi | = |Qi | − 1 =
ki
∑
|ρj | − 1 ≤ λ(ki + 1) − 1,
j=0
which is to say that
dProjΓ γ (H0 , Hi ) ≤ λdΓX (H0 , Hi ) + λ − 1.
Since λ ≥ 1, we have
1
dΓX (H0 , Hi ) − λ + 1 ≤ dΓX (H0 , Hi ) ≤ dProjΓ γ (H0 , Hi ),
λ
where the latter inequality holds because ProjΓ γ ⊆ ΓX. For any 0 ≤
m ≤ n, we can repeat the same argument, using the geodesic segment
Qm,n = γ([m − 12 , n − 12 ]), without changing the value of λ, and obtain
125
the same inequalities. Indeed, Qm,n ⊂ Qn , and hence the disc diagram
argument proceeds exactly as before. Thus the inclusion ProjΓ γ ,→ ΓX is a
quasi-isometric embedding: for all m, n ≥ 0,
1
dΓX (Hm , Hn ) − λ + 1 ≤ dProjΓ γ (Hm , Hn ) ≤ λdΓX (Hm , Hn ) + λ − 1.
λ
Since Λγ properly contains, and has the same vertex-set as, ProjΓ γ, it follows
that the inclusion Λγ ,→ ΓX is also a quasi-isometric embedding. Thus, if
λ < ∞, then statement (1) of the trichotomy holds.
The case λ = ∞: In this case, for each n ≥ 0, there exists i = i(n)
such that λi(n) > n. Thus there exists j ≤ i such that |ρj | > n. Let Wi = Uj
be the corresponding hyperplane in the path ψi , so that ρj → N (Wi ) has
length m > n. Let K1 , K2 , . . . Km be the dual curves in Di emanating from ρj ,
numbered increasingly according to their positions on γ; this is unambiguous
since no two of these dual curves cross.
Let Q′i be the subtended part of Qi , i.e. the smallest connected subpath
of Qi containing each 1-cube c of Qi such that c is dual to Ka for some
a ∈ {1, 2, . . . , m}. Note that |Q′i | ≥ m.
Let K̂1 and K̂m be paths in Di on the carriers of K1 and Km respectively
such that K̂1 , K̂m contain no 1-cubes dual to K1 , Km and map to geodesic
segments in X. Choose these paths so that K̂1 , K̂m , ρj and Q′i bound a
subdiagram Ei ⊂ Di , as shown at left in Figure 6–6. Suppose that Ei has
minimal area in the sense that any hexagon move in Di that could push a
square from Ei into Di − Ei has already been made, so that no two dual
curves in Ei emanating from K̂1 or from K̂m cross.
Each dual curve Kp in Ei emanating from ρj ends on Q′i , for otherwise
Kp would cross K1 or Km , which is impossible by the earlier argument using
minimality of Di . Hence, if L is a dual curve traveling from K̂1 to K̂m , then
L crosses each Kp . If L1 and L2 are dual curves emanating from K̂1 , then L1
and L2 do not cross. First, suppose that no dual curve L travels from K̂1 to
K̂m .
126
Figure 6–6: The diagram Ei is shown at left inside Di , and at right with the
dual curves named in the proof of Theorem 6.1.1.
The dual curves in Ei can thus be partitioned into three disjoint sets,
H1 , Hm , which respectively consist of the dual curves emanating from K̂1 and
K̂m and ending on Q′i , and K, which consists of dual curves traveling from Q′i
to ρj .
Let ha = |Ha | for a ∈ {1, m} and let h = min(h1 , hm ). For 1 ≤ p ≤ h1 , let
L1p be the pth dual curve to cross K̂1 , counting from Q′i to ρj , and similarly, let
th
Lm
dual curve to cross K̂m , counting from ρj to Q′i . See the picture
p be the p
of Ei at right in Figure 6–6.
Finding bicliques: Consider the subdiagram Fi1 bounded by K̂1 , a
geodesic path L̂1h1 in the carrier of L1h1 , and the subtended path Q′i,1 ⊂ Q′i , as
shown at right in Figure 6–6. Since dual curves in Fi1 travel from L̂1h1 to Q′i,1
or from K̂1 to Q′i,1 , we see that Fi1 is a h1 × |L1h1 | flat triangle. On the other
hand, the analogously-defined subdiagram Fim is a h1 × |Lm
1 | flat triangle. We
thus have an embedding
K ha
2
,
|La |
b
2
,→ Λ(γ),
where a ∈ {1, m}. Let t1 = |L1h1 |, let tm = |Lm
1 |, and let d < m be the
1
number of dual curves traveling from ρj to Q′i that separate Lm
1 from Lh1 .
Then t1 + tm = m − d. Hence, if d is bounded as m increases, at least one of
t1 , tm is unbounded.
Verifying (3) when max(h1 , hm ) is uniformly bounded: Let c ∈ Q′i
be a 0-cube, so that at most h1 + hm dual curves in Ei separate c from ρi .
127
Hence, since the map Ei1 → X1 does not increase distances, we see that Q′i lies
in Nh1 +hm +1 (Wi ). Suppose that max(h1 , hm ) ≤ ϵ for some uniform constant
ϵ. Then for each n ≥ 0, the geodesic ray γ has a subpath, namely Q′i , with
|Q′i | ≥ m > n, such that Q′i ⊂ N2ϵ+1 (Wi ), verifying statement (3) of the
trichotomy, with R = 2ϵ + 1.
Verifying (3) when max(t1 , tm ) is bounded, d unbounded, h1
1
unbounded: The length-d subpath of Q′i between Lm
1 and Lh1 cannot
be separated from ρj by any dual curve, and hence maps to a subpath of
γ ∩ Nt1 +tm (Wi ) of length d. Thus (3) holds if d is unbounded, with R = 1.
Verifying (3) when t1 is unbounded and h1 is bounded: Suppose
that h1 ≤ R. Then Q′i,1 is a subpath of γ of length t1 + R that is separated
from Wi by at most h1 dual curves. If such an R < ∞ can be chosen
independently of m, and t1 is unbounded in m, then (3) holds.
Verifying (2): The remaining possibility is that, say, h1 is unbounded in
m and t1 is unbounded in m. Since K h1 , t1 ,→ Λ(γ), the graph Λ(γ) does not
2
2
have thin bicliques, i.e. (2) holds.
Finally, if some dual curve L in Ei travels from K̂1 to K̂m , then simply
replace Wi with the hyperplane to which L maps and argue in the same way.
Indeed, the subdiagram Ei′ of Ei between L and Q′i has the same structure
as Ei , and we either locate a large biclique in Λ(Q′i ) or conclude that Q′i
fellow-travels with L.
This concludes the proof of the first assertion: either γ quasi-isometrically
embeds in the contact graph, or γ fellow-travels with hyperplanes for arbitrarily large amounts of time, or the full subgraph of the contact graph generated
by the projection of γ does not have thin bicliques.
We now prove the second assertion. Let v ⊂ ∂△ X be the simplex
represented by W(γ). If dim v ≥ 1, then γ bounds a fractional flat, by
Lemma 5.2.7. Hence suppose that v is a 0-simplex. Since {Hi }i≥0 is a UBS,
128
by Lemma 5.1.2 and local finiteness of X, there exists a subsequence (ij )j≥0
such that, for all j ≥ 1, the hyperplanes Hij−1 and Hij+1 are separated by Hij .
Since dim v ≥ 0, all but finitely many Hi have the property that Hi crosses
only finitely many of the Hij .
Note that any path in ΓX from Hi0 to Hij either passes through Hij−1
or through some hyperplane H crossing Hij−1 . Hence, if γ has bounded
projection, then there exists K ≥ 0 such that for all k > K, there is
a hyperplane Vk that crosses HiK and Hik , and thus that crosses Hij for
K ≤ j ≤ k. Thus either γ has unbounded projection, or there exists a
hyperplane V such that V ⊥Hij for all j ≥ K, by Lemma 2.1.1. Then since
each Hi crosses only finitely many of the Hij , the hyperplane V must cross all
but finitely many of the hyperplanes Hi dual to 1-cubes of γ.
On the other hand, let γN = γ([N − 21 , ∞)). Then every hyperplane W
separating γN from V must cross all but finitely many of the Hij . Let SN be
the set of such hyperplanes, and note that SN ⊂ SN +1 for all N ≥ 0 since
∪
γN +1 ⊂ γN . If S = N SN is infinite, then S and W(γ) form a weak nearjoin
in ∆X, whence γ bounds a fractional flat by Lemma 5.2.7. If S is finite, then
by convexity of carriers, γ fellow-travels with V with constant |S|.
We conclude that, if diamΓX (ProjΓ γ) < ∞ and γ does not fellow-travel
with a hyperplane, then the simplex v ⊂ ∂△ X represented by W(γ) has
dimension at least 1. By Lemma 5.2.7, γ bounds a fractional flat.
Theorem 6.1.1 immediately yields the following “preliminary” characterization of locally finite CAT(0) cube complexes with unbounded contact
graphs, in terms of isolated 0-simplices in the boundary.
Corollary 6.1.3. Let X be locally finite. Then the contact graph ΓX has
finite diameter only if every isolated 0-simplex v ∈ ∂△ X lies in the image of
the injection ∂△ H → ∂△ X induced by the inclusion H ,→ X.
Remark 6.1.4. Since N (H) is a convex subcomplex of X, Theorem 5.1.7
guarantees that ∂△ N (H) is a subcomplex of ∂△ X. But N (H) ∼
= H × I, where
129
I is an interval, and hence, by Proposition 5.1.10, there is an isomorphism
∂△ H ∼
= ∂△ N (H). The map ∂△ H → ∂△ X is obtained by composing this
isomorphism with the simplicial embedding ∂△ N (H) → ∂△ X.
Proof of Corollary 6.1.3. Suppose that diam(ΓX) < ∞. Let v ∈ ∂△ X
be an isolated 0-simplex. Then by Theorem 5.2.2, there exists a geodesic
ray γ → X such that W(γ) is a UBS representing γ, since v is maximal
and therefore visible. Since ΓX has finite diameter, γ must have bounded
projection, and hence, by Theorem 6.1.1, we may suppose that γ fellow-travels
with a hyperplane H. Indeed, γ cannot bound a fractional flat, for otherwise
v would lie in a 1-simplex of ∂△ X. Since γ fellow-travels with H, all but
finitely many hyperplanes V ∈ W(γ) cross H, and hence there is a UBS V,
almost-equivalent to W(γ), such that for all V ∈ V, the hyperplane V crosses
H. The inclusion H ,→ X induces a simplicial embedding ∂△ H → ∂△ X,
by Theorem 5.1.7. Moreover, the set {H ∩ V | V ∈ V} is a UBS in H
representing a 0-simplex v ′ ∈ ∂△ H, and the induced embedding ∂△ H → ∂△ X
sends v ′ to v. Hence, if diam(ΓX) < ∞, then every isolated 0-simplex of X lies
in the image of the simplicial boundary of some hyperplane.
6.2
Characterizing bounded contact graphs
In proving Theorem 6.2.3, it is convenient to work in the crossing graph.
Fix a base hyperplane H0 . Recall the definition of the grade g(H) of the
hyperplane H with respect to H0 . First, g(H0 ) = 0. If for some n ≥ 1, the
hyperplane H crosses some hyperplane H ′ with g(H ′ ) = n − 1 and H does
not cross any hyperplane of grade n − 2, then g(H) = n. Let Wn be the set
of hyperplanes of grade at most n. This coincides with the notion of grading
in Remark 3.1.3, but we are using the crossing graph instead of the contact
graph.
Lemma 6.2.1. Let X be strongly locally finite and compactly indecomposable.
Then for any finite subgraph Λ of ∆X, the complement ∆X − Λ is connected.
130
Proof. Proof that ∆X is connected: First, ∆X is connected. Otherwise,
X is the wedge sum of two proper subcomplexes, and is thus not compactly
indecomposable. Indeed, let H, H ′ be hyperplanes lying in different components of the crossing graph. If H ′′ separates H from H ′ , then H ′′ cannot lie
in the same component of ∆X as both H and H ′ , so we may choose these
hyperplanes so that H ⌣
⊥ H ′ . Now N (H) ∩ N (H ′ ) consists of a single 0-cube
x, since, if N (H) ∩ N (H ′ ) contained distinct 0-cubes x, y, each hyperplane
separating x from y would cross H and H ′ .
Now, neither of H, H ′ is finite, since X is compactly indecomposable.
Thus we can choose 0-cubes y ∈ N (H) and y ′ ∈ N (H ′ ) so that x, y, y ′ are
pairwise-distinct. Suppose by way of a contradiction that P → X is a path
joining y to y ′ so that x ̸∈ P . Let A → N (H) be a shortest path joining x
to y and let B → N (H ′ ) be a shortest path joining y ′ to x, and consider a
minimal-area disc diagram D → X bounded by AP B. See Figure 6–7. Let
K be the dual curve emanating from the initial 1-cube of A and let K ′ be the
dual curve emanating from the terminal 1-cube of B. Let U be the hyperplane
to which K maps, and let U ′ be the hyperplane to which K ′ maps. If U = U ′ ,
then H⊥U ⊥H ′ , a contradiction. If K crosses K ′ , then H⊥U ⊥U ′ ⊥H ′ , another
contradiction. By minimality of area, K and K ′ end on P . Let P ′ be the
subpath of P between and including the 1-cubes c and c′ dual to K and K ′ , as
in Figure 6–7, so that P ′ = cQc′ , where Q is a path such that |Q| < |P | and
x ̸∈ Q. On the other hand, x ∈ N (U ) ∩ N (U ′ ), and no hyperplane W crosses
U and U ′ , since otherwise the path H⊥U ⊥W ⊥U ′ ⊥H ′ would join H and H ′
in ∆X. Thus we can replace H and H ′ by U and U ′ respectively, and replace
P by Q, and obtain a lower-length counterexample. Eventually, we are in a
situation where |Q| = 0 and x ̸∈ Q. But then Q ∈ N (U ) ∩ N (U ′ ) is separated
from x ∈ N (U ) ∩ N (U ′ ), and thus U and U ′ are crossed by a common
hyperplane, a contradiction. Hence no such P can exist, i.e. every path from
N (H) to N (H ′ ) passes through x. It is now obvious that x disconnects X.
131
Figure 6–7: If ∆X is disconnected, then X has a cut-0-cube.
Disconnecting ∆X in a restriction quotient: Now suppose Λ is
a finite graph that separates ∆X. Let W ′ ⊂ W be the set of hyperplanes
that do not correspond to vertices of Λ. Let X′ = X(W ′ ) be the restriction
quotient corresponding to this subset, and let q : X → X′ be the associated
restriction quotient. Now, if W, W ′ are hyperplanes of X, then W, W ′ cross
if and only if q(W ) crosses q(W ′ ) in X′ . In other words, ∆X′ ∼
= ∆X − Λ. If
∆X − Λ is disconnected, it follows that ∆X′ is disconnected, and therefore
that X′ ∼
= Y1 ∪{a} Y2 , where Yi is a nonempty subcomplex containing at least
one hyperplane, and a is a single 0-cube. Let K = q −1 (Y1 ) ∩ q −1 (Y2 ) = q −1 (a).
Since q is continuous, K separates X into two nonempty subcomplexes.
On the other hand, if b, b′ ∈ K are 0-cubes, then q(b) = q(a) = q(b′ ),
so that b(H) ̸= b(H ′ ) if and only if H is a hyperplane corresponding to
a vertex of Λ. Thus K is finite and disconnects X, contradicting compact
indecomposability.
Lemma 6.2.2. Let X be compactly indecomposable and have finite degree. If
diam(ΓX) < ∞, then diam(∆X) < ∞.
Proof. Since X is compactly indecomposable, ∆X is path-connected, by
Lemma 6.2.1. If H and H ′ are osculating hyperplanes and d∆X (H, H ′ ) = R,
then a simple disc diagram argument shows that there is a shortest path
H = H0 ⊥H1 ⊥ . . . ⊥HR−1 ⊥HR = H ′ in ∆X with the property that Hi ⌣H
⊥ j for
all i, j, and hence N (H) ∩ N (H ′ ) contains a 0-cube of degree at least R + 1.
132
Thus, if H, H ′ osculate, then d∆X (H, H ′ ) ≤ D − 1, where D is the degree of
X. Thus d∆X (H, H ′ ) ≤ D whenever H ⌣
⊥ H ′ , and it follows from the triangle
inequality that diam(∆X) ≤ (D − 1) diam(ΓX). See Figure 6–8.
Figure 6–8: The red and green hyperplane osculate but the shortest path
joining them in ∆X has length 7. A disc diagram argument shows that the
intersection of their carriers is a vertex of degree at least 8.
We now prove the main theorem of this chapter, characterizing locally
finite, leafless, one-ended CAT(0) cube complexes with bounded contact
graph.
Theorem 6.2.3. Let X be a strongly locally finite, essential, compactly
indecomposable CAT(0) cube complex. Then
diam(ΓX) ≤ diam(∆X) ≤ diam(∂△ X) ≤ 2 diam(∆X) − 2 ≤ 2D diam(ΓX) − 2,
where D is the degree of X, and we have the following:
1. If ∂△ X is bounded, then ΓX is bounded and X does not contain a
rank-one geodesic ray.
2. X decomposes as an iterated pseudoproduct if and only if ∆X is
bounded.
3. If X decomposes as an iterated pseudoproduct, then ∂△ X is bounded.
4. If D < ∞, then (3) and (4) hold with ∆X replaced by ΓX.
Remark 6.2.4. The above inequalities are sharp. If X ∼
= R2 , then
diam(ΓX) = 2 diam(ΓX) − 2 = diam ∂△ X = 2.
For i ≥ 1, let Fi be a non-diagonal quarter-flat, bounded by rays Ii
and Oi . Let X1 = F1 and for i ≥ 2, construct Xi by attaching Fi to Xi−1
133
′
by identifying the ray Ii with the sub-ray Oi−1
⊂ Oi−1 beginning at the
second 0-cube. Xi is locally finite and fully visible. For each i < ∞, the
simplicial boundary ∂△ Xi is a subdivided line segment of length i, and thus
has diameter i. It is easily seen that diam(∆Xi ) = i + 1. However, Xi is
not leafless for any i. To rectify this, attach a non-diagonal quarter-flat F0
to Xi by identifying one of the bounding rays of F0 with I1 in the obvious
cubical way; the resulting cube complex X satisfies diam(∂△ X) = i + 1 and
diam(ΓX) = i + 1. Note that the spiral of quarter-flats S obtained by gluing
∪
two copies of i≥1 Xi along I0 has an unbounded contact graph, and ∂△ S
consists of R together with a pair of isolated 0-simplices corresponding to a
“spiraling” geodesic ray. See Figure 6–9.
Figure 6–9: An infinite spiral of quarter-flats. The red hyperplane is at distance 5 from the blue one in ∆X, and the simplicial boundary of the part of
X pictured is a subdivides interval of length 5. The arrowed geodesic spends
finite time in each quarter-flat, and represents a pair of isolated simplices in
∂△ X.
Proof of Theorem 6.2.3. By Lemma 2.2.12, X is compactly indecomposable.
Assertion (4) follows from the claimed inequality and Lemma 6.2.2.
∆X bounded when ∂△ X bounded: This follows from the claimed
inequality, which we establish below.
134
No rank-one ray when diam(∂△ X) < ∞: Suppose ∂△ X is bounded.
Then either ∂△ X consists of a single 0-simplex, or every 0-simplex lies in a
1-simplex. In the latter case, every geodesic ray bounds a fractional flat by
Lemma 5.2.7. In the former case, there is a single 0-simplex v ∈ ∂△ X and
hence, for each geodesic ray γ, all but finitely many hyperplanes of X cross γ.
This contradicts leaflessness.
Obtaining a pseudoproduct when ∆X is bounded: Let ∆X be
bounded, so that there exists R ≤ diam ∆X < ∞ and a hyperplane H0 such
that, for all H ∈ W, the grade g(H) of H in ∆X, with respect to H0 , is at
most R. By leaflessness, we can assume R ≥ 2, for otherwise H0 crosses each
H ̸= H0 , and thus H0 is a leaf. Moreover, choose R so that there exists H
with g(H) = R.
Let V1 be the set of hyperplanes H with 0 ≤ g(H) ≤ R − 1, and let V2
be the set of grade-R hyperplanes. Since X is compactly indecomposable and
H0 is not a leaf, N (H0 ) cannot be compact, and thus there are infinitely many
hyperplanes crossing H0 . Hence V1 is infinite.
On the other hand, let HR be a grade-R hyperplane. Then since R ≥ 2,
the hyperplanes HR and H0 do not cross. Since HR is not a leaf, there exists
a hyperplane H ′ such that HR separates H ′ from H0 . Any path in ∆X from
H ′ to H0 contains HR or some hyperplane crossing HR , and thus H ′ has grade
at least R, whence, since every hyperplane has grade at most R, we have
H ′ ∈ V2 . Now H ′ is not a leaf, and thus separates some H ′′ from H0 . Thus H ′′
has grade R, by the same argument. Hence V2 is infinite.
The quotients X → Qi : For i ∈ {1, 2}, let Qi be the cube complex dual
to the wallspace (X0 , Vi ) whose walls are those induced by the hyperplanes in
Vi . This gives a cubical quotient qi : X → Qi ; this is just a restriction quotient
that collapses the hyperplanes in W − Vi .
Let Vi be the set of hyperplanes in Qi , so that there is a natural bijection
Vi → Vi given by V 7→ qi (V ). Note that, if V, V ′ ∈ Vi are hyperplanes, then
V ⊥V ′ if and only if qi (V )⊥qi (V ′ ). Moreover, if V, V ′ osculate, then qi (V )
135
and qi (V ′ ) osculate. For V, V ′ ∈ V1 , the converse is true. If no hyperplane
separates q1 (V ) from q1 (V ′ ), then any hyperplane U separating V, V ′ must
belong to V2 . But if g(V ), g(V ′ ) ≤ R − 1 and U separates V, V ′ , then
g(U ) ≤ R − 1, a contradiction. In other words, V1 is inseparable, and hence
V, V ′ ∈ V1 contact if and only if their images in Q1 do.
However, it could happen that there exist non-contacting V, V ′ ∈ V2
such that q2 (V ) osculates with q2 (V ′ ), if every hyperplane separating V, V ′ has
grade at most R − 1.
The embedding e : X → Q1 × Q2 : Fix a base 0-cube xo ∈ X.
Recall that this is a section xo : W → W ± of the map π : W ± → W that
associates each halfspace of X to the corresponding hyperplane. Moreover, xo
is automatically consistent and canonical.
For any base 0-cube (yo1 , yo2 ) ∈ Q1 × Q2 . Then yo1 is a consistent, canonical
orientation of the hyperplanes q1 (V ) ∈ V1 and yo2 a consistent canonical
orientation of the hyperplanes in V2 . Define yoi (qi (V )) = qi (xo (V )), i.e. the
image in Qi of the halfspace associated to V that contains xo . This is the
halfspace of Qi containing qi (xo ). In other words, let yoi = qi (xo ).
We now define a map X0 ∋ x 7→ ψ(x) = (ψ 1 (x), ψ 2 (x)) ∈ Q1 × Q2
as follows. Let ψ i (x)(qi (V )) = qi (x(V )). Then, since x is consistent, for all
V, V ′ ∈ Vi , we have x(V )∩x(V ′ ) ̸= ∅ and hence qi (x(V ))∩qi (x(V ′ )) ⊇ qi (x(V )∩
x′ (V )) ̸= ∅. Thus ψ i (x) is a consistent orientation of the hyperplanes of Qi .
Since x differs from xo on finitely many hyperplanes in Vi , and ψ i (x)(qi (V )) ̸=
ψ i (xo )(qi (V )) only if x(V ) ̸= xo (V ), the 0-cube ψ i (x) is canonical. Hence ψ(x)
is a genuine 0-cube of Q1 × Q2 .
Now ψ i (x)(qi (V )) differs from ψ i (y)(qi (V )) when qi (x(V )) and qi (y(V ))
are different halfspaces associated to qi (V ). This happens if and only if
x(V ) ̸= y(V ), so that dQ1 ×Q2 (ψ(x), ψ(y)) counts the hyperplanes in V1 ⊔ V2 =
W on which x, y differ. Thus ψ : X0 → (Q1 × Q2 )0 is an isometric embedding,
and hence the map ψ extends to an isometric embedding e : X → Q1 × Q2 ,
136
since each median graph uniquely determines a CAT(0) cube complex whose
1-skeleton is the given median graph.
The embedding a : Q1 → X: Choose a base 0-cube xo ∈ N (H0 ).
Let Y0 be the set of 0-cubes y in X such that every hyperplane separating
y from xo belongs to V1 . Let Y ⊂ X be the convex hull of Y0 . Each 0-cube
y ∈ Y, viewed as a section of π, has the property that y(W ) = xo (W )
for W ∈ V2 , since W cannot separate y from xo . In other words, Y is
isomorphic to the restriction quotient obtained by restricting each 0-cube of
X to V1 ! More precisely, for each q(x) ∈ Q1 , define a(q(x)) : W → W ±
by a(q(x))(W ) = x(W ) for W ∈ V1 and, for all W ∈ V2 , let a(q(x))(W )
be the halfspace containing H0 (and thus xo ). This is well-defined since no
W ∈ W2 crosses H0 . This orientation is consistent, since x is consistent on W1 .
If a(q(x)) orients W ∈ W1 inconsistently with V ∈ W2 , then V separates W
from H0 , but we have seen this to be impossible.
Finally, a(q(x)) agrees with xo on all but finitely many hyperplanes,
namely those hyperplanes in V1 on which x, xo disagree. Hence a(q(x)) is
canonical and belongs to Y, and, if x ∈ Y, then a(q(x)) = x. In the usual
way, one checks that q(x) 7→ a(q(x)) determines an isometric embedding
a : Q1 → X.
Note that, in general, we do no obtain an isometric embedding Q2 → X.
Indeed, since there are in general pairs of hyperplanes in V2 that are separated
by a hyperplane in V1 , there is no natural place X to send the basepoint of
Q1 !
The convex hull of e(X): Let Z ⊆ Q1 × Q2 be the convex hull of e(X).
Every hyperplane of Q1 × Q2 is of the form q1 (V ) × Q2 or Q1 × q2 (H), where
V ∈ V1 and H ∈ V2 , and each hyperplane of the former type crosses each
hyperplane of the latter type. It follows that every hyperplane of Q1 × Q2
crosses e(X), whence Z = Q1 × Q2 by Proposition 2.2.6.
The 2-cubes: Let H ∈ V2 be a grade-R hyperplane. By leaflessness,
there exists an infinite set U0 , U1 , . . . of grade-R hyperplanes such that H
137
separates Ui from H0 , for all i ≥ 0, and for all i ≥ 1, the hyperplanes Ui+1
and Ui−1 are separated by Ui . Every Ui crosses a set {Wij }j of grade-R − 1
hyperplanes, each of which must cross H. Let Ω be the full subgraph of ∆X
generated by {Wij } ∪ {H}. Then every path in ∆X from Ui to H0 passes
through H or through one of the Wij , since H separates Ui from H0 . Thus
∆X − Ω has at least two nonempty components, namely the component
containing U0 and the component containing H0 . Hence, by Lemma 6.2.1, Ω is
infinite, whence infinitely many grade-(R − 1) hyperplanes cross H in X.
Induction: Now Q1 has infinitely many hyperplanes and ∆Q1 has finite
diameter – grading from H0 , every hyperplane has grade at most R − 1. If
R = 2, then Q1 ∼
= H0 × [− 12 , 12 ]. Otherwise, by leaflessness, the set V21 of
grade (R − 1) hyperplanes is infinite, and, letting V11 be the (infinite) set of
hyperplanes of grade at most R − 2, we proceed as before, to find that
Q1 ∼
= Q11 Q12 .
Continuing in this way, since the highest grade decreases at each step, we find
that, as long as R > 2, we can continue the pseudo-product decompositions,
∼
using the above argument. This terminates with QR−1
= H0 × [− 12 , 21 ] ∼
=
1
N (H0 ). Hence, if diam(∆X) < ∞, then X decomposes as an iterated
pseudoproduct.
Conversely, suppose that X is an iterated pseudoproduct. Then, for each
hyperplane H ̸= H0 of X, either there exists a maximal m ≤ k − 1 such that H
survives in the restriction quotient X → Q01 → Q11 → . . . → Qm
1 , or H crosses
∼ m+1 Q2 , so that, by
a hyperplane crossing Q01 . In the latter case, Qm
1 = Q1
m+1
the definition of a pseudo-product, H crosses a hyperplane that crosses Qm+1
.
1
Hence H is at distance at most k + 1 from H0 in ∆X, and diam(∆X) < ∞.
We have proved (2).
Proof of (3): Suppose that X ∼
= Q1 Q2 , where Q1 is the restriction
quotient obtained from the set V1 of hyperplanes of grade at most R − 1,
and X2 is the restriction quotient obtained from the set V2 of hyperplanes
138
of grade R, where R ≤ diam(ΓX) = diam(∆X) < ∞. We shall prove that
diam(∂△ X) ≤ 2R − 2, establishing one of the claimed inequalities and thus
proving that ∂△ X has finite diameter.
Since a(Q1 ) ⊂ X is convex, we have a simplicial embedding ∂△ a :
∂△ Q1 → ∂△ X, by Theorem 5.1.7. Let A1 = ∂△ a(∂△ X) be its image. Hence, by
Proposition 5.1.10, we have
∂△ (Q1 × Q2 ) ∼
= A1 ⋆ ∂△ Q2 ,
since the isometric embedding e ◦ a : Q1 → Q2 embeds Q1 convexly in Q1 × Q2
(it embeds as Q1 × yo2 ).
Let A2 ∼
= ∂△ Q2 , so that ∂△ (Q1 ×Q2 ) = A1 ⋆A2 . In particular, A1 ∩A2 = ∅.
Thus every simplex of ∂△ (Q1 × Q2 ) is of the form
[u0 , u1 , . . . , up , v0 , . . . , vq ],
where each ui represented in ∂△ X by a unidirectional boundary set Ui ⊂ V1
and each vj is represented by a unidirectional boundary set q2 (Wj ), where
Wj ⊂ V2 is a UBS.
Let Q2 be a UBS of hyperplanes in Q2 that represents the 0-simplex q.
Let {Wi }i≥0 ⊆ Q2 be a set of hyperplanes such that for all i ≥ 1, the hyperplane Wi separates Wi−1 from Wi+1 . By the pseudoproduct decomposition, for
each i ≥ 0, there exist infinitely many hyperplanes H of grade at most R − 1
that cross Wi in X, and thus cross Wj in X, for all j < i. Hence there is a
UBS Q1 representing a 0-simplex q 1 ∈ ∂△ Q1 such that [q 1 , q] is a 1-simplex of
im(∂△ X → ∂△ (Q1 × Q2 )). In other words, each 0-simplex of ∂△ Q2 is adjacent
in the image of ∂△ X to some 0-simplex of ∂△ Q1 .
We now argue by induction on R. First suppose that R ≥ 3 and, by
induction, that there is a function K : N → N such that, for all 0-simplices
s, s′ ∈ ∂△ Q1 , there is a path in ∂△ X of length at most K(R − 1) joining s to
s′ . Then, if q, q ′ ∈ ∂△ Q2 are 0-simplices, they are respectively at distance 1
from some 0-simplices s, s′ ∈ ∂△ Q1 , by the preceding paragraph, and hence
139
there is a path in ∂△ X of length at most K(R) = K(R − 1) + 2 joining q, q ′ .
Hence we have
diam(∂△ X) ≤ K(R) = K(2) + 2(R − 2).
Next, let R = 2 and let s1 , s2 be 0-simplices of ∂△ Q1 , so that for
i ∈ {1, 2}, the simplex si is represented by a UBS S i consisting only of grade-1
hyperplanes. Let {Sji }j≥0 ⊆ S i be a maximal set of pairwise non-crossing
hyperplanes. Either [s1 , s2 ] is a 1-simplex, and we are done, or we can choose
S i so that S 1 ∩ S 2 = ∅.
In the latter case, for all j, k ≥ 0, there is a geodesic path
Sj1 ⊥H0 ⊥Sk2
in ∆X, where H0 is the base hyperplane of the grading. Thus the set of
hyperplanes H such that H⊥Sj1 and H⊥Sk2 is nonempty. On the other hand,
the set of such H, together with {Sj1 , Sk2 }, generates a subgraph Λ ⊂ ∆X such
that ∆X − Λ has at least two infinite components, since X is leafless. Thus
there are infinitely many such H, and hence there is a minimal UBS U such
that, for all U ∈ U, we have a path
Sj1 ⊥U ⊥Sk2
in ∆X. But then s1 and s2 are both adjacent to the 0-simplex u represented
by U. Thus K(2) = 2 and
diam(∂△ X) ≤ 2R − 2 ≤ 2 diam(∆X) − 2,
as required.
The remaining inequality: It remains to show that diam(∆X) ≤
diam(∂△ X). Choose R ≤ diam(∆X). Also, assume that we have chosen
R ≥ 2, for if we can not make this choice then essentiality is contradicted.
Choose hyperplanes U0 and V0 such that d∆X (U0 , V0 ) = R ≥ 2. By
essentiality, there exists a minimal UBS U and a minimal UBS V such that
140
U0 ∈ U and V0 ∈ V are initial hyperplanes in the given UBSes: for all U ′ ∈ U,
U0 separates U ′ from V0 , and for all V ′ ∈ V, V0 separates V ′ from U0 .
Let u, v be the 0-simplices at infinity represented by U and V respectively.
If u, v lie in different components of ∂△ X, then diam(∂△ X) = ∞ and we are
done.
Therefore, let u, s1 , . . . , sk , v be a path in (∂△ X)1 joining u to v.
Let Si be a UBS representing the 0-simplex si . Then there is a path
σ = U ⊥S1 ⊥ . . . ⊥Sk ⊥V in ∆X, where U ∈ U, V ∈ V and for each 1 ≤ t ≤ k,
we have St ∈ St . Without loss of generality, U0 and V0 both separate U
from V . Hence σ must either contain U0 (respectively, V0 ) or σ must contain
a hyperplane St that crosses U0 (respectively, a hyperplane Sr that crosses
V0 ). Hence, in the worst case, we have a path U0 ⊥St ⊥St+1 ⊥ . . . ⊥Sr ⊥V0 of
length r − t + 2 joining U0 to V0 . Thus k + 1 ≥ r − t + 2 ≥ R, whence
diam(∂△ X) ≥ diam(∆X).
6.3
Rank-rigidity from the contact graph viewpoint
Let X be a finite-dimensional, locally finite CAT(0) cube complex on
which G acts leaflessly and properly. We shall give a modified proof of the
rank-rigidity theorem of Caprace and Sageev, showing either that G contains
a rank-one isometry that does not virtually stabilize a hyperplane, or X
decomposes as the product of two convex subcomplexes, provided either G
acts cocompactly or there is no fixed point in ∂∞ X. We also show that this
dichotomy corresponds exactly to the dichotomy between unboundedness
and boundedness of ΓX, in the cocompact case, and to connectedness or
disconnectedness of ∂△ X in the event that there is no fixed point at infinity.
Theorem 6.1.1 also allows us to say slightly more about rank-one isometries:
they have quasi-geodesic axes in the contact graph, or their axes fellow-travel
with hyperplanes. Our version of the rank-rigidity theorem states that G
contains a rank-one element of the former type, in the cocompact case.
Our strategy is as follows. First, we use one of the tools introduced
by Caprace-Sageev, the double-skewering lemma, to show that, if ΓX has
141
infinite diameter, then ΓX contains an unbounded orbit, and thus G contains
a rank-one isometry. We then use leaflessness to prove an auxiliary result,
saying that if ΓX is bounded, then X is one-ended (in the cocompact case).
We then show, that if ΓX is bounded, then ∆X has diameter 2, by applying
double-skewering to the contact graph. Originally, we applied Theorem 6.2.3
at this point, to show that the pseudo-product decomposition must in fact
be a product in this situation. However, this step in the proof amounts to
essentially the same argument as is used to prove the irreducibility criterion
in [CS11]. Hence we simply use this criterion, which is stated in terms of the
strongly-separated hyperplanes of Behrstock and Charney, noting that two
hyperplanes are strongly-separated exactly when their distance in the crossing
graph exceeds 2. In the proper, cocompact case, the use of the irreducibility
criterion in [CS11] is justified by a somewhat complicated (although very nice)
decomposition theorem. We also use this result, but in a slightly different way,
relying on the auxiliary lemma about one-endedness. A detailed discussion of
the similarities and differences between the statement and proof given here,
and that given in [CS11], appears before Theorem 6.3.6.
First, we note a direct consequence of Theorem 6.1.1 that classifies
isometries of X in terms of their action on ΓX.
Corollary 6.3.1. Let G act on the locally finite, finite-dimensional CAT(0)
cube complex X and let g ∈ G. If g acts on ΓX with an unbounded orbit, then
g is rank-one.
Let g ∈ G be a combinatorially rank-one element. Suppose that for
all n > 0, and for all hyperplanes H, g n ̸∈ Stab(H). Then there exists a
quasi-isometric embedding α : R → ΓX such that gα = α.
Proof. If g stabilizes a cube, then g stabilizes a finite family of hyperplanes,
and hence has no unbounded orbit in ΓX. Likewise, if g exchanges the
halfspaces of some hyperplane, then g has a fixed point in ΓX. Hence, if g has
an unbounded orbit in ΓX, then g is combinatorially hyperbolic, by Haglund’s
classification of combinatorial isometries [Hag07].
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Suppose g is not rank-one. Then for any combinatorial axis γ for g, and
for any hyperplane H crossing γ, we have dΓX (g n H, H) ≤ 3 for all n ∈ Z.
Indeed, g n H and H cross a common isometric half-flat in X. The contact
graph of a diagonal half-flat has diameter 3, and that of a non-diagonal
half flat has diameter 2. If U is any other hyperplane, then dΓX (g n U, U ) ≤
3 + 2dΓX (H, U ), and thus every g-orbit in ΓX is bounded. This proves the first
assertion.
To prove the second assertion, let γ : R → X be a combinatorial geodesic
axis for the rank-one element g. For each n ≥ 0, let γn = γ([−n, ∞)), and let
Hp be the hyperplane dual to γ([p, p + 1]) for each p ∈ Z.
Suppose that there exist λ ≥ 1, µ ≥ 0 such that, for all p, q ≥ 0,
λ−1 |p − q| − µ ≤ dΓX (Hp , Hq ) ≤ λ|p − q| + µ.
Then for all p′ , q ′ ∈ Z, there exists m ∈ Z such that g m Hp′ and g m Hq′
cross γ0 , in 1-cubes [p, p + 1] and [q, q + 1] with |p − q| = |p′ − q ′ |. Since
dΓX (Hp′ , Hq′ ) = dΓX (Hp , Hq ), we have
λ−1 |p′ − q ′ | − µ ≤ dΓX (Hp′ , Hq′ ) ≤ λ|p′ − q ′ | + µ,
and hence α(γ) is a g-invariant quasi-geodesic in ΓX.
If no such λ, µ exist, then γ0 satisfies conclusion (2) or (3) of Theorem 6.1.1. Suppose (3) holds, so that there exists R ≥ 0 such that, for all
p ≥ 0, there exists a hyperplane H such that γ0 contains a subpath σp of
length at least p with σp ⊆ NR (H). Translating σp and H by some g m , we find
that for all p ≥ 0, there is a subpath σp′ ⊂ γ such and a hyperplane H p such
that
p p
γ([− , ]) ⊆ σp′ ⊂ NR (H p ).
2 2
Now γ(0) is contained in the R-neighborhood of finitely many hyperplanes,
since X is locally finite. Therefore, by König’s lemma, there exists a hyperplane H such that γ ⊂ NR (H). Again by local finiteness, there are only N
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such hyperplanes, for some N < ∞, and these are permuted by g. Hence
g n (H) = H for some 0 < |n| ≤ N , a contradiction.
The remaining possibility is that (2) holds for γ0 , so that the set {Hn }n≥0
generates a subgraph of ΓX that contains Kp,p for all p ≥ 0. Hence, for all
p ≥ 0, there exist m, n ∈ Z with |m − n| ≥ p such that Hm and Hn contact.
Translating by an appropriate power of g, we may assume that m ≤
−p
2
and
n ≥ p2 .
For m < r < n, Hr crosses Hm or Hn . Thus the subgraph of
ΓX generated by {Hn }n∈Z has diameter at most 3, and in particular,
diamΓX (ProjΓ (γ0 )) < ∞. By Theorem 6.1.1, either γ0 lies in a uniform
neighborhood of a hyperplane or γ0 bounds an isometrically embedded eighthflat S ⊆ X. In the former case, ⟨g⟩ virtually stabilizes a hyperplane, as above.
In the latter case, since X is locally finite, ⟨g⟩S contains a half-flat bounded
by γ, contradicting the fact that g is rank-one.
6.3.1
Caprace-Sageev double-skewering from a contact graph
viewpoint
The goal of this section is to prove:
Lemma 6.3.2. Let G act properly, cocompactly, and leaflessly on X. If ΓX is
unbounded, then there exists a rank-one element g ∈ G that does not virtually
stabilize a hyperplane.
Proof. For each N > 0, there exists a hyperplane V and a hyperbolic isometry
g ∈ G such that some axis γ of g crosses V and gV and dΓX (V, gV ) > N . This
follows from Lemma 6.3.3, since ΓX is unbounded. If γ bounds a half-flat,
then diamΓX (W(γ)) ≤ 3, so by choosing N > 3, we ensure that γ is rankone. Note that if γ fellow-travels with a hyperplane H, then H crosses every
element of W(γ), so that the same choice of N also guarantees that g does not
virtually stabilize a hyperplane.
The following two lemmas support Lemma 6.3.2.
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Lemma 6.3.3. If 2 ≤ N < diam(ΓX), then there exists a hyperbolic isometry
g ∈ G, a hyperplane H, and n ∈ Z such that dΓX (g n H, H) ≥ N .
Proof. Let U, V be hyperplanes such that dΓX (U, V ) ≥ N . Since N ≥ 2, the
hyperplanes U, V cannot cross. By Lemma 6.3.4, there exists a hyperbolic
isometry g ∈ G such that some axis γ for g crosses U and V and contains
points arbitrarily far from U, V in each halfspace. Since G acts with finitely
many orbits of hyperplanes, there exists H ∈ W(γ) and n ∈ Z such that U
and V separate H from g n H. Hence dΓX (H, g n H) ≥ dΓX (U, V ) ≥ N .
The following lemma, due to Caprace and Sageev, is a fundamental
tool allowing one to find hyperbolic isometries, given a pair of non-crossing
hyperplanes. It is proved in greater generality in [CS11]: the conclusion holds
if cocompactness if dropped, provided X is locally finite and G does not fix a
point on the visual boundary. We shall use this form below, translating the
hypotheses about fixed points on the visual boundary into a statement about
G-invariant simplices of ∂△ X.
Lemma 6.3.4 (Double-skewering [CS11]). Let G act properly and cocompactly
on X. Let U, V be non-crossing distinct hyperplanes. Then there exists a
hyperbolic element g ∈ G such that some axis γ of g crosses U, V and contains
points arbitrarily far from U (and from V ) in each halfspace. g can be chosen
so that U separates V from gV .
6.3.2
One-endedness when ΓX is bounded
We need the following in order to use Theorem 6.2.3.
Lemma 6.3.5. Let G act properly, cocompactly, and leaflessly on X. If X has
more than one end, then G contains a rank-one isometry. If ΓX is bounded,
then X is one-ended.
Proof. Suppose that X has more than one end. Then there exists a compact,
convex subcomplex K ⊂ X such that X − K has a finite collection C1 , . . . Ck
of unbounded components, with k ≥ 2. Given g ∈ G and subcomplexes A, B ⊂
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X, we say that gK separates A, B if A and B lie in distinct components of
X − gK.
Since G acts properly, there exists g ∈ G such that gK ∩ K = ∅. Hence
some hyperplane H separates gK from K. By leaflessness, there exists a
hyperplane U in the same halfspace of H as K. Since H is not a leaf, we
can choose U so that K separates H from U . By Lemma 6.3.4, there exists
a hyperbolic h ∈ H such that some axis γ for h crosses H and U , and thus
crosses K. If γ bounds a half-flat F, then no two points of γ are separated
by the removal of a compact subcomplex. But, since γ crosses K and K is a
compact separating subcomplex in X, γ cannot bound a half-flat. Thus h is a
rank-one isometry of X.
By Corollary 6.3.1, either h has a quasigeodesic axis in ΓX, which is
therefore unbounded, or h virtually stabilizes a hyperplane H crossing K
and hK. If every rank-one isometry of X is of the latter form, then there
exists a finite set of hyperplanes, {H1 , . . . , Hk }, such that for all g ∈ G, the
subcomplexes K, gK are both crossed by Hi for some 1 ≤ i ≤ k with g n ∈ GHi
for some n ∈ Z. It follows that a single hyperplane H crosses every translate
of K, and thus that every point of X lies in a uniform neighborhood of H,
contradicting leaflessness. Hence ΓX is unbounded.
To see this, suppose that Hi crosses K and gK. Let hK be a translate
of K such that gK separates K and hK. Suppose that Hi does not cross
hK. Then some Hj crosses K and hK, and thus crosses gK. Hence a single
hyperplane H crosses each gK. Now, by cocompactness, there exists R < ∞
such that X = NR (GK). Thus, for all 0-cubes x ∈ X, we have dX (x, N (H)) ≤
R + diam(K), contradicting essentiality.
6.3.3
An alternative proof of the rank-rigidity theorem
We now give a slightly modified roof of the rank-rigidity theorem of
Caprace-Sageev, and it is worth comparing the statement and proof to that
in [CS11]. The statement differs in the fact that the two alternatives given
146
by the theorem – either X is a product, or G contains a rank-one isometry,
are shown to correspond to the dichotomy: either ΓX is unbounded or
diam(ΓX) = 2.
We proceed as follows: if ΓX is unbounded, then the above lemmas,
which rely on double-skewering, show that G contains a rank-one isometry
that does not virtually stabilize a hyperplane. If ΓX is bounded, we show
that X does not contain any strongly-separated hyperplanes, in the sense
of [BC11]. It then follows from the irreducibility criterion in [CS11] that
X decomposes as a product. In other words, our proof of Theorem 6.3.6
differs from that in [CS11] in that we replace Lemma 6.2 of that paper by
the simpler Lemma 6.3.2. We still rely on the irreducibility criterion using
strongly-separated walls and on the double-skewering lemma. Lemma 6.2
of [CS11] says that, if U, V are a pair of non-strongly-separated hyperplanes
skewered by an axis γ of a rank-one element g, then g is contracting and, in
particular, rank-one. The proof of this result relies on the fact that sufficiently
long geodesic segments that are hyperplane-equivalent to segments in γ are
contained in a uniform neighborhood of γ. Compare this to Theorem 6.1.1,
which says something similar even when γ is not periodic.
When we are assuming a proper, cocompact, leafless action of G on X,
our application of the irreducibility criterion is slightly different from that
of Caprace-Sageev. They show that either the irreducibility criterion – X is
a product if there are no strongly-separated walls – applies, or X has only
compact hyperplanes. They then invoke the flat plane theorem to show that
every hyperbolic element is rank-one. We invoke their decomposition result
(Corollary 4.9 of [CS11]) in a slightly different way, appealing to Lemma 6.3.5
– and thus to the Double-Skewering Lemma – rather than bringing into play
the flat plane theorem.
We will apply the irreducibility criterion when X has bounded contact
graph. In this situation, G can be assumed not to fix a point on ∂∞ X. Indeed,
Lemma 6.3.5 implies that X has no compact hyperplane. Now Corollary 4.9
147
of [CS11] guarantees that X ∼
= X1 × . . . × Xp × Y, where a finite-index
subgroup G′ ≤ G acts on Y with no fixed point on the boundary, and every
hyperplane of every Xi is compact. Now, the claimed product decomposition
in Theorem 6.3.6 exists unless Z or Y is bounded. If Z is bounded, then we
apply the proof of Theorem 6.3.6 to Y and find either a rank-one element
of G′ or a decomposition of Y into unbounded factors, and we are done.
Hence suppose that Y is bounded. Then Z is unbounded, and hence X1 is
unbounded (say). Then, letting Z1 = X2 × . . . × Xp × Y, we see that Z1 must
be unbounded, since X contains hyperplanes of the form H × Z1 , where H
is a compact hyperplane of X1 . By Lemma 6.3.5, Z1 is unbounded. We may
therefore assume that G fixes no point on the visual boundary of X, and we
thus have access to the irreducibility criterion.
Theorem 6.3.6 (Rank-rigidity, cocompact contact graph form). Let G
act properly, cocompactly, and leaflessly on X. If ΓX is unbounded, then
G contains a rank-one isometry of X that does not virtually stabilize a
hyperplane. If ΓX is bounded then there exist unbounded convex subcomplexes
Q1 , Q2 ⊂ X such that X ∼
= Q1 × Q2 .
Proof. If ΓX is unbounded, then G contains the desired rank-one isometry
by Lemma 6.3.2. Hence suppose that ΓX has finite diameter (so that ∆X
has finite diameter, by cocompactness and Lemma 6.2.2). Since G acts
properly and cocompactly, X is locally finite. Since G acts leaflessly, X is
leafless. Since G acts properly, leaflessly, and cocompactly, we may assume
that X is one-ended, by Lemma 6.3.5. Hence, by Theorem 6.2.3, X ∼
=
(Q0 × [− 21 , 12 ]) Q2 . . . QR , where R < ∞. Also, since X is one-ended, X
decomposes as a product provided it has no strongly-separated hyperplanes;
this is the irreducibility criterion, and its use is justified by Lemma 6.3.5 and
the discussion preceding this proof.
Let H0 ∼
= Q0 be the base hyperplane, and let R ≥ 2 be the maximal
grade in ∆X, with respect to H0 . We first establish, using cocompactness and
leaflessness, that R = 2.
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The maximal grade is exactly 2: If R = 1, then every hyperplane
crosses H0 , contradicting leaflessness. Hence suppose that diam(∆X) = R ≥
2, and let U be a grade-R hyperplane. Then by Lemma 6.3.4, we have a
hyperbolic isometry g and a hyperplane H such that gH and H are separated
by H0 and U . Hence d∆X (gH, H) = R. On the other hand, gH separates
H from g 2 H, and d∆X (gH, g 2 H) = R, so that, since any path in ∆X from
g 2 H to H either contains H or some hyperplane U that crosses H, we have
d∆X (g 2 H, H) ≥ 2R − 2. Hence 2R − 2 ≤ R, i.e. R ≤ 2. Thus ∆X has diameter
at most 2, and we can choose H0 so that the maximal grade is exactly 2.
Applying the irreducibility criterion: If U, V ∈ W, then since
d∆X (U, V ) ≤ 2, the hyperplanes U, V are not strongly separated in the sense
of Behrstock-Charney [BC11]. It now follows from Proposition 5.1 of [CS11]
that, if V1 is the set of grade-1 hyperplanes, and V2 is the set of grade-2 or
grade-0 hyperplanes, and V ∈ V1 , H ∈ V2 , then V ⊥H, i.e. X ∼
= Q1 × Q2 ,
where Q1 ∼
= H0 and Q2 is a convex subcomplex crossed by V2 . Since X is
one-ended, we can argue as in the proof of Theorem 6.2.3 to show that V1 and
V2 are infinite, so that Q1 and Q2 are unbounded.
Just as we restated the proper, cocompact rank-rigidity theorem in terms
of the contact graph, we state the other version given in [CS11] in terms of the
simplicial boundary. In this version, we still rely on the irreducibility criterion
for part of the proof, but for part of the proof we can turn to Theorem 6.2.3
and an argument like that used in Lemma 6.3.5.
Lemma 6.3.7. Let G act on the strongly locally finite CAT(0) cube complex
X. If G fixes a point of ∂∞ X, then ∂△ X contains a G-invariant simplex. G
has a finite-index subgroup that fixes a simplex of ∂△ X.
Proof. Let β be a CAT(0) geodesic ray such that gβ fellow-travels with β for
all g ∈ G. Let g ∈ G. Then W(β)△W(gβ) is finite. Indeed, if H crosses β but
does not cross gβ, then H separates β(t) from gβ(t) for all sufficiently large t,
149
and there are thus only finitely many such H. It is easily seen that W(β) is a
UBS. Hence the simplex v ⊂ ∂△ X represented by W(β) is G-invariant.
By Theorem 5.1.8, v dimension d < ∞. Since G stabilizes v, we have a
homomorphism G → Sd whose kernel fixes v. Thus G virtually fixes v.
Theorem 6.3.8 (Rank-rigidity, simplicial boundary form). Let G act properly
and leaflessly on the locally finite CAT(0) cube complex X. Suppose that ∂△ X
contains no 0-simplex fixed by a finite-index subgroup of G. Then (∂△ X)1
has infinite diameter if and only if G contains a rank-one isometry of X.
Otherwise, ∂△ X decomposes as a join of two nonempty subcomplexes.
Proof. X is leafless and locally finite. By Lemma 6.3.7, G does not fix a point
on ∂∞ X. We therefore have access to the Double-Skewering lemma, which was
shown in [CS11] to hold under these conditions. Hence, if ΓX is unbounded,
then G has a rank-one isometry g that does not virtually stabilize a hyperplane, by Lemma 6.3.2. Thus there are isolated 0-simplices v, v ′ of ∂△ X, by
Corollary 6.1.3, and thus (∂△ X)1 has infinite diameter. If ΓX is bounded,
then the grading argument in the proof of Theorem 6.3.6 shows that X has
no strongly-separated hyperplanes, so that by the irreducibility criterion,
X∼
= Q1 × Q2 , as desired. By Proposition 5.1.10, we have ∂△ X ∼
= ∂△ Q1 ⋆ ∂△ Q2 .
Finally, if (∂△ X)1 has infinite diameter, then it does not decompose as a
nontrivial join, from which it follows that X cannot decompose as a product,
by Proposition 5.1.10. Thus either G contains the desired rank-one isometry, or ΓX is bounded. If X is one-ended, then Theorem 6.2.3 ensures that
diam((∂△ X)1 ) ≤ 2 diam(∆X) − 2, so that the crossing graph cannot be
bounded.
The remaining situation is therefore that in which X has bounded contact
graph and unbounded simplicial boundary, and X is not one-ended. Let K
be a compact separating subcomplex, and let gK be disjoint from K; such
a gK exists by properness of the action. Then gK can be chosen so that
some hyperplane H separates gK from K. Using double-skewering, we find a
150
hyperbolic element h with an axis γ that passes through K and conclude that
h is rank-one, as in the proof of Lemma 6.3.5.
151
CHAPTER 7
The combinatorial Tits boundary and divergence
Earlier, we discussed the visual boundary of a CAT(0) space M , and saw
that a disadvantage of this boundary is that it fails to distinguish between
quite different metric spaces; the visual boundaries of both the hyperbolic and
Euclidean planes are homeomorphic to S 1 , for example. This issue is partly
addressed by introducing the Tits boundary, whose construction is described
in detail in, e.g. [BH99].
Roughly, the Tits boundary of ∂T M is obtained by metrizing the visual
boundary as follows. Given two geodesic rays γ, γ ′ in M , with common
basepoint, one defines an angle θ(γ, γ ′ ) using a Euclidean comparison angle;
the angle between two equivalence classes of rays is defined by taking the
supremum of the angle between representatives with common initial point as
this point varies. This yields a metric on ∂∞ M taking values in [0, π]. The
Tits metric is the associated path metric. ∂T M is the visual boundary of M ,
equipped with the Tits metric. If M is a complete CAT(0) space, then ∂T M is
a complete CAT(1) geodesic space.
The Tits boundary has many features analogous to those of ∂△ X. For
example, if M is Gromov-hyperbolic, then ∂T M is a discrete space [BH99].
Moreover, if M has the right “geodesic extension property”, then ∂T M
decomposes as a spherical join if and only if M decomposes as the product of
unbounded convex subcomplexes (at least when M is locally compact) [BH99];
compare this fact to Proposition 5.1.10.
We also saw that, if u, v ⊂ ∂△ X are disjoint simplices, then u, v represent
the two ends of a bi-infinite combinatorial geodesic in X. The first of the
following properties of ∂T M , when M is a complete proper CAT(0) space,
described in [BH99], is analogous to this “visibility” property:
152
1. If γ, γ ′ are geodesic rays representing points at Tits distance more than
π, then the union γ ∪ γ ′ fellow-travels with a bi-infinite geodesic.
2. If γ ∪ γ ′ does not fellow-travel with a bi-infinite geodesic, then the
corresponding points on the Tits boundary are at distance equal to the
angle distance, and are joined by a geodesic in ∂T M .
3. If γ is a bi-infinite geodesic in M , then γ represents two points of ∂T M
at distance at least π, with equality if and only if γ bounds a Euclidean
half-plane in M .
4. ∂T M has diameter π if and only if there is no rank-one bi-infinite
geodesic in M .
In this chapter, we describe a simple pseudo-metric on ∂△ X, where X
is a strongly locally finite CAT(0) cube complex. In Section 7.2, we define a
naive version of the divergence function of a pair of combinatorial geodesic
rays in X, and relate this to ∂△ X. More precisely, we show that the simplices
of ∂△ X represented by the combinatorial geodesic rays γ, γ ′ lie in the same
component of ∂△ X if and only if the associated divergence function is at
most linear. In Section 7.3, we bring this notion of divergence into line
with the more robust, quasi-isometry invariant notion of divergence (see
e.g. [Ger94b, Ger94a, DMS10, KL98, BC11]). This yields a result about the
divergence and the divergence of geodesics in groups acting properly and
cocompactly on geodesically complete cube complexes: both are linear if and
only if the contact graph of the associated cube complex decomposes as a
nontrivial join, and both are otherwise at least quadratic. This generalizes a
result of Behrstock-Charney for right-angled Artin groups, which is stated in
terms of join decompositions of the generating graph [BC11].
From now on, the notation for paths changes slightly, for simplicity.
Whereas, before, we regarded a path P : I → X as a cubical map of CAT(0)
cube complexes, so that 0-cubes had the form P (− 21 + n) for n ∈ Z, we now
only need consider the images of 0-cubes. In the interest of simpler notation,
we say a geodesic ray γ is a map γ : N ∪ {0} → X0 such that, for each
153
hyperplane U , there is at most one s ∈ N such that U separates γ(s) from
γ(s − 1). Given xo ∈ X0 , we say that γ emanates from x0 if γ(0) = x0 .
7.1
∂△ X as a combinatorial Tits boundary
Let X be a strongly locally finite CAT(0) cube complex with simplicial
boundary ∂△ X. A simplex-path in ∂△ X is a sequence
P = v0 , v 1 , . . . , v n
of simplices, such that vi ∩ vi+1 is a common face of vi and vi+1 for each
0 ≤ i ≤ n − 1. The length |P |s of P is equal to n, i.e. to the number of
faces vi ∩ vi+1 , counted with multiplicity. Let u, v ⊂ ∂△ X be simplices. Then
η(u, v) = inf P |P |s , where the infimum is taken over all simplex-paths P
such that u ⊆ v0 and v ⊆ vn . Hence η(u, v) = ∞ if and only if u and v
lie in different components of X, and η(v, v ′ ) = 0 if and only if v, v ′ lie in a
common simplex. The map η : Simplices(∂△ X)2 → R≥0 ∪ {∞} is clearly a
pseudo-metric on the set Simplices(∂△ X) of ∂△ X. Note, moreover, that if v, v ′
are 0-simplices, then
d∂△ X (v, v ′ ) − 2 ≤ η(v, v ′ ) ≤ d∂△ X (v, v ′ ).
Now, if γ, γ ′ : R+ → X are combinatorial geodesic rays, then denote by
v, v ′ the simplices respectively represented by γ, γ ′ . Let η(γ, γ ′ ) = η(v, v ′ ).
This defines a pseudo-metric on the space of all combinatorial geodesic rays,
and η(γ, γ ′ ) = 0 if are γ, γ ′ are consumed by a common geodesic ray.
7.2
Divergence of pairs of combinatorial geodesic rays
Let X be fully visible. Fix a base 0-cube xo ∈ X. For each integer r ≥ 0,
∪
let Lr be the set of 0-cubes x ∈ X with dX (xo , x) = r. Let Br = ri=0 Lr . A
combinatorial path P → X is r-avoiding if P ∩ Br−1 = ∅, i.e. if every 0-cube
of P is at distance at least r from xo .
154
If γ, γ ′ are combinatorial geodesic rays emanating from xo , we define their
divergence function div(γ, γ ′ ) : Z≥0 → R by
div(γ, γ ′ )(r) = inf |P |,
P
where the infimum is taken over all r-avoiding combinatorial paths P joining
γ(r) ∈ Lr to γ ′ (r) ∈ Lr .
Given a function f : N → R, we say that X has f -divergence (e.g. linear,
quadratic) if for all xo ∈ X0 and all pairs of rays γ, γ ′ emanating from xo , we
have, for all sufficiently large r ≥ 0, that
div(γ, γ ′ )
≤N
f (r)
for some fixed constant N < ∞, independent of xo , γ, γ ′ . If there is no such
N , then X has super-f divergence. This leads to a slightly weaker notion of,
say, linear divergence than is used when using the notion of divergence due to
Gersten; this difference is addressed below. We will study the divergence of
geodesic rays using the simplicial boundary. At present, I do not understand
the relationship between divergence and asymptotic cones well enough to
say if our approach is somehow analogous to that taken in [DMS10] (maybe
disconnectedness of the simplicial boundary corresponds somehow to cutpoints in the asymptotic cone); nor is it clear what the asymptotic cones of
median graphs look like, in general. Both of these issues are intriguing.
7.2.1
Bounding the divergence from above
Lemma 7.2.1. If γ, γ ′ , γ ′′ are combinatorial geodesic rays emanating from xo ,
then for all r ≥ 0,
div(γ, γ ′ )(r) ≤ div(γ ′ , γ ′′ )(r) + div(γ, γ ′′ )(r),
i.e. div(−, −) satisfies a triangle inequality.
155
Proof. Let Pr be an r-avoiding path joining γ(r) to γ ′′ (r) and let Qr be an ravoiding path joining γ ′′ (r) to γ ′ (r). Then Pr Qr is an r-avoiding path joining
γ(r) to γ ′ (r).
Lemma 7.2.2. Let X be locally finite. Let γ, γ ′ be geodesic rays emanating
from the basepoint xo and representing simplices v, v ′ ⊂ ∂△ X respectively. The
divergence of γ, γ ′ satisfies:
1. If v ⊆ v ′ , then div(γ, γ ′ )(r) ≤ 2r + B for all r ≥ 0 and some B ∈ R.
2. If v, v ′ belong to a common simplex, then div(γ, γ ′ )(r) ≤ 2r + B for all
r ≥ 0 and some B ∈ R.
Proof. Let V, V ′ be the sets of hyperplanes dual to γ, γ ′ respectively. Let
H = V − V ′ and let H′ = V ′ − V.
Proof of (1): If v ⊆ v ′ , then H is finite. We establish two claims, first
treating the case in which H = ∅, and then treating the general case.
Claim I: If H = ∅, i.e. if V ⊆ V ′ , then
div(γ, γ ′ )(r) = dX (γ(r), γ ′ (r)) ≤ 2r
for all r ≥ 0.
Proof of Claim I. Fix r ≥ 0. Consider the set of hyperplanes crossing the
geodesic segment Pr = γ([0, r]). Since H = ∅, this set of hyperplanes is
partitioned into two subsets: Ar is the set of hyperplanes that cross Pr and
Pr′ = γ ′ ([0, r]). The set of remaining hyperplanes, Br , all cross Pr and γ ′ − Pr′ ,
since every hyperplane that crosses γ also crosses γ ′ .
The set of hyperplanes crossing Pr′ consists of Ar , together with a set Cr
of hyperplanes U such that each U crosses γ − Pr , or separates an infinite
subray of γ from an infinite subray of γ ′ , and thus crosses all but finitely many
elements of H. See Figure 7–1, and note that |Ar | + |Br | = |Ar | + |Cr | = r.
Note also that if B ∈ Br and C ∈ Cr , then B⊥C, so that the hyperplanes in
Br are orientable independently of those in Cr .
156
Figure 7–1: The sets Ar , Br , Cr are shown at left. At right is the path Qr .
We now construct an r-avoiding geodesic segment Qr joining x′r = γ ′ (r)
to xr = γ(r), from which the claimed equality follows. Let Q′r (0) = x′r .
Let B0 , . . . , Bb be the hyperplanes in Br , numbered so that x′r ∈ N (B0 )
and B0 ⌣
⊥ B1 ⌣
⊥ ... ⌣
⊥ Bb . This assumption is justified, since any
hyperplane separating Bi from Bi+1 must cross Pr and thus belong to Br .
Any hyperplane separating x′r from B0 is likewise in Br . For 0 ≤ t ≤ b, let
Q′r (t) orient all hyperplanes U ∈ W − Br toward the halfspace x′r (U ), and
let Q′r (t)(Bs ) = x′r (Bs ) for s > t. For s ≤ t, let Q′r (t)(Bs ) = X − x′r (Bs ).
This creates a geodesic segment from x′r to Q′r (t) that crosses exactly the set
Br of hyperplanes. Note that each Q′r (t) is separated from xo by Ar ∪ Cr , and
thus dX (Q′r (t), xo ) ≥ r for all t. In the same manner, we flip the hyperplanes
Cr successively to yield a geodesic segment Q′′r from Q′r (b) to xr , with Q′′r
separated from xo by Ar ∪ Br . The segment Qr = Q′r Q′′r is the desired
r-avoiding geodesic segment.
Claim II: If H is finite, then there exists a geodesic ray γ ′′ emanating from xo
such that dX (γ ′′ (r), γ ′ (r)) ≤ 2|H| for all r ≥ 0.
Proof of Claim II. Let Ar , Br , Cr be defined as in the proof of Claim I. Let
Hr be the set of H ∈ H that cross Pr . Since H is finite, Hr has uniformly
bounded cardinality h. Let H = {H0 , . . . , Hh }, where for i < j, the hyperplane
157
Hi separates the 1-cube N (Hj ) ∩ γ from xo . Let R be the subpath of γ
joining xo to the 0-cube of N (Hh ) ∩ γ separated from xo by Hh . Let α be the
combinatorial geodesic subray of γ emanating from the terminal 0-cube of
R. Then Hh crosses every hyperplane dual to α and α → N (Hh ). For t ≥ 0,
define α′ (t) to be the 0-cube that orients each hyperplane U toward α(t)(U ),
except for U = Hh , in which case α′ (t)(Hh ) = X − α(t)(Hh ). This orientation
is consistent and canonical for each t, and the map t 7→ α′ (t) yields a geodesic
ray α′ → N (Hh ) emanating from the penultimate 0-cube of R. Concatenating
α′ with the first |R| − 1 1-cubes of R yields a geodesic γh′′ → X such that γh′′ (t)
is separated from γ(t + 1) by at most one hyperplane, namely Hh , for each
t ≥ 0. The claim now follows by induction on |H|. Note that by Claim I, for
all r ≥ 0,
div(γ, γ ′′ )(r) ≤ 2|H|.
By Claim I, for all r ≥ 0,
div(γ ′ , γ ′′ )(r) = dX (γ ′ (r), γ ′′ (r)) ≤ 2r
and
div(γ, γ ′′ )(r) = dX (γ(r), γ ′ (r)) ≤ 2|H|,
so that assertion (1) follows from Lemma 7.2.1 with B = 2|H|.
Proof of (2): Let v, v ′ belong to the common simplex v ′′ . If v ⊆ v ′ ,
or vice versa, then assertion (2) reduces to assertion (1). There are now two
possibilities. First, we could have that v, v ′ are distinct 0-simplices belonging
to a common 1-simplex v ′′ . Second, if v ̸⊂ v ′ and v ′ ̸⊂ v, and v has positive
dimension, then there exists a simplex u = v ∩ v ′ that is properly contained in
both v, v ′ .
Let V, V ′ be as before. Then V = V0 ⊔ H and V ′ = V0′ ⊔ H′ , where H, H′
are finite, and V0 , V0′ satisfy the following conditions:
158
1. If U ∈ V0 , then either U crosses both γ and γ, or U crosses every
element of V0′ .
2. If U ′ ∈ V0 , then either U ′ crosses both γ and γ, or U crosses every
element of V0 .
In either case (v ∩ v ′ = u or v ∩ v ′ = ∅), the proof of assertion (1) shows,
mutatis mutandis, that
div(γ, γ ′ )(r) ≤ dX (γ(r), γ ′ (r)) + 2(|H| + |H′ |)
for all r ≥ 0.
We can now relate the divergence function to the “combinatorial Tits
distance” η defined above.
Proposition 7.2.3. Let X be strongly locally finite and fully visible. Let γ, γ ′
be geodesic rays emanating from the basepoint xo and representing simplices
v, v ′ ⊂ ∂△ X respectively. There exists a constant A ∈ R, depending only on
η(v, v ′ ), such that for all r ≥ 0 and some B ∈ R,
div(γ, γ ′ )(r) ≤ 2Ar + B.
Hence, if v, v ′ lie in the same component of ∂△ X, then div(γ, γ)(r) is at most
linear.
Proof. If η(v, v ′ ) = ∞, there is nothing to prove; A = η(v, v ′ ) suffices. Hence
suppose η(v, v ′ ) = N < ∞. We shall argue by induction on N . The claim
holds for N = 0, with A = 1, by Lemma 7.2.2. Indeed, if N = 0, then v, v ′ lie
in a common simplex.
Suppose that the claim holds for some N − 1 ≥ 0. Let γ, γ ′ satisfy
η(γ, γ ′ ) = N . Then there exist simplices u0 , uN ⊂ ∂△ X such that v ⊆ u0 , v ′ ⊆
uN , and there is a minimum-length simplex-path
P = u0 , u 1 , . . . , u N
159
in ∂△ X joining u0 to uN . Let γ ′′ be a geodesic ray emanating from xo and
representing uN ∩ uN −1 . By induction, div(γ, γ ′′ )(r) ≤ 2Ar + B1 for all r and
some fixed A depending on N , since η(v, uN ∩ uN −1 ) = N − 1.
Since uN ∩ uN −1 and v ′ lie in a common simplex, namely uN , Lemma 7.2.2
implies that
div(γ ′ , γ ′′ )(r) ≤ 2r + B1 ,
so that by Lemma 7.2.1, we have
div(γ, γ ′ )(r) ≤ 2(A + 1)r + B1 + B2 .
Since A = 1 suffices for N = 0, we may take A = N + 1 and B = B1 + B2 , and
conclude that div(γ, γ ′ )(r) ≤ 2r (η(γ, γ ′ ) + 1) + B.
Remark 7.2.4. In Theorem 7.3.6, we will need control over B as γ, γ ′
vary. In that situation, we will need to prove that, if (∂△ X)1 has diameter
A − 1 < ∞, then for any choice of xo , γ, γ ′ , we have
div(γ, γ ′ )(r) ≤ 2Ar + B
for some uniform constant B. In general, Proposition 7.2.3 gives no such
bound on B. We therefore need the following lemma about divergence of
cubical products.
Lemma 7.2.5. Let X ∼
= Q1 × Q2 , where Q1 and Q2 are unbounded convex
subcomplexes of the strongly locally finite CAT(0) cube complex X. Then
there exists A ∈ R such that for all xo ∈ X, all pairs γ, γ ′ of geodesic rays
emanating from xo ,
div(γ, γ ′ )(r) ≤ 2Ar
for all r ≥ 0.
Proof. Note that, in a fractional flat, the divergence of any two rays with
common basepoint is bounded above by 4r. If γ ∪ γ ′ is a bi-infinite geodesic,
then the claim follows from this fact and Lemma 5.2.8, with A = 2.
160
More generally, let Qi be the Qi -factor containing xo . If γ → Q1
and γ ′ → Q2 , then γ × γ ′ is a non-diagonal quarter-flat in X, so that
div(γ, γ ′ )(r) ≤ 2r. If γ, γ ′ → Q1 , then since there is a geodesic ray α → Q2
emanating from xo , we have a subcomplex F = (γ ∪ γ ′ ) × α in X that is
the union of the non-diagonal quarter-flats γ × α, γ ′ × α, each of which is
isometrically embedded in X. There is thus an r-avoiding path of length 4r
joining γ(r) to γ ′ (r).
Let γi , γi′ denote the projections of γ, γ ′ to Qi , for i ∈ {1, 2}. Without
loss of generality, γ1 is a combinatorial geodesic ray. Now every hyperplane
crossing γ1 crosses γ, and if U ∈ W(γ1 ) and V ∈ W(γ) − W(γ1 ), then U ⊥V .
Hence, by the proof of Lemma 7.2.2,
div(γ1 , γ)(r) = dX (γ(r), γ1 (r))
for all r ≥ 0.
If γ2′ is an infinite ray, then
div(γ2′ , γ1 )(r) = 2r
for all r, since those rays co-bound a non-diagonal quarter-flat, so that
div(γ, γ ′ )(r) ≤ 2r + div(γ1 , γ)(r) + div(γ2′ , γ ′ )(r) ≤ 6r
by Lemma 7.2.1.
If γ1′ is an infinite ray, then it was shown above that div(γ1 , γ1′ )(r) ≤ 4r,
so that
div(γ, γ ′ )(r) ≤ div(γ1 , γ1′ )(r) + div(γ ′ , γ1′ )(r) + div(γ, γ1 )(r)
≤ 4r + div(γ, γ1 )(r) + div(γ ′ , γ1′ )(r)
≤ 8r
161
7.2.2
Bounding the divergence from below
Let Sr be a shortest r-avoiding path joining γ(r) to γ ′ (r). Then we
can decompose Sr as a concatenation of geodesic segments in the following
“minimal” way. First, let P1r be the longest subpath of Sr whose initial 0-cube
is γ(r) and which crosses each hyperplane at most once. If Sr − P1r ̸= ∅, let P2r
be the longest subpath of Sr − P1r that is a geodesic segment in X and begins
at the terminal 0-cube of P1r . Continuing in this way, we find that
Sr = P1r P2r . . . Pcrr ,
where each Pir is a geodesic segment in X − Lr . Moreover, for each i ≥ 2, the
r
, for otherwise we could
hyperplane dual to the initial 1-cube of Pir crosses Pi−1
r
have extended Pi−1
geodesically. This is the fan decomposition of Sr , shown in
Figure 7–2.
Figure 7–2: A heuristic picture of the fan decomposition of a shortest ravoiding path from γ(r) to γ ′ (r).
Suppose that γ, γ ′ have the property that, for all r ≥ 0, the path Sr
is a geodesic segment of X, i.e. Sr = P1r . Then the simplices v, v ′ ⊂ ∂△ X
represented by γ, γ ′ either lie in a common simplex of ∂△ X or satisfy v ∩ v ′ ̸=
∅.
162
Indeed, suppose that v ∩ v ′ = ∅, so that the set U = W(γ) ∩ W(γ ′ )
is finite. By finitely folding and truncating, without affecting either the
simplices v, v ′ or the geodesics P1r , for r sufficiently large, we may assume
that U = ∅. Then every hyperplane separating γ(r) from xo separates γ ′ (r)
from γ(r), and the same is true of each hyperplane separating γ(r) from
xo . Hence |P1r | = div(γ, γ ′ )(r) = 2r for all r ≥ 0. (Or, more generally,
2r ≥ div(γ, γ ′ )(r) ≥ 2r − 2|U|, if we didn’t fold and truncate.)
Let P1r = QQ′ be the concatenation of two geodesic segments of length r.
Let Cr , Cr′ be the initial length-r segments of γ, γ ′ respectively. Let D → X
be a minimal-area disc diagram with boundary path Cr QQ′ (Cr′ )−1 . Without
changing the endpoints of Cr , Cr′ , or modifying the sets W(γ), W(γ ′ ), we may
choose Cr and Cr′ so that no two dual curves in D emanating from Cr cross,
and no two dual curves emanating from Cr′ cross.
Since any two distinct dual curves in D have a total of 4 ends, and ∂p D is
the concatenation of 3 geodesic segments, dual curves in D, each dual curve in
D maps to a distinct hyperplane, and thus D → X is an isometric embedding.
Every dual curve in D travels from Cr to QQ′ or from Cr′ to QQ′ , since U = ∅
and QQ′ = P1r is a geodesic segment.
Since every 0-cube of QQ′ is separated in D from xo by at least r dual
curves, it follows that every dual curve emanating from Cr crosses every
dual curve emanating from Cr′ , and hence, by Lemma 2.1.1, the rays γ and
γ ′ can be chosen within their almost-equivalence classes to lie in a common
isometrically embedded quarter-flat. In particular, v, v ′ belong to a common
simplex. This motivates the following lemma:
Lemma 7.2.6. Let γ, γ ′ be combinatorial geodesic rays emanating from xo
and representing 0-simplices v, v ′ ∈ ∂△ X respectively. For each r ≥ 0, let Sr
be a shortest r-avoiding path joining γ(r) to γ ′ (r). Suppose that there exists
K such that, for all r ≥ 0, |Sr | − dX (γ(r), γ ′ (r)) ≤ 2K. Then v, v ′ lie in a
common simplex, i.e. η(v, v ′ ) = 0. If v, v ′ are arbitrary simplices, and the
same hypothesis holds, then η(v, v ′ ) ≤ 1.
163
Proof. The preceding discussion proves this in the case K = 0. For K ≥ 1, the
hypothesis says that for all r ≥ 0, there are at most K hyperplanes that cross
Sr in more than one 1-cube. As before, assume that v ∩ v ′ = ∅, for otherwise
η(v, v ′ ) ≤ 1 by definition. Hence, by folding and truncating finitely many
times, we may assume that W(γ) ∩ W(γ ′ ) = ∅.
Suppose that η(v, v ′ ) > 1, which is to say that v and v are disjoint and do
not belong to any common simplex. Let Hr be the hyperplane dual to the rth
1-cube of γ, and define Hr′ analogously for γ ′ . The there exists R, N ≥ 0 such
that, for all r ≥ R, the hyperplane Hr does not cross HN′ . Also, we can choose
R such that, for all r ≥ 0, there are at most R hyperplanes U that cross Hr
and Hr′ .
Let Sr be a shortest r-avoiding path joining γ(r) to γ ′ (r). Each 0-cube
c ∈ Sr is separated from xo by at least r hyperplanes. Let Qr , Q′r be the
initial, length-r segments of γ, γ ′ respectively, emanating from xo . Let Dr be
a minimal-area disc diagram bounded by Qr Sr (Q′r )−1 . Since γ and γ ′ have
no common dual hyperplanes, every dual curve in Dr travels from Qr or Q′r
to Sr . Without modifying xo , γ(r), γ ′ (r) or v, v ′ , we may change Qr , Q′r so
that no two dual curves in Dr emanating from Qr cross, and no two dual
curves emanating from Q′r cross. See Figure 7–3. Let ur be the number of
dual curves in Dr that start and end on Sr , so that |Sr | = 2r + 2ur .
Figure 7–3: The diagram Dr showing that either η(v, v ′ ) ≤ 1 or ur → ∞.
164
Let c ∈ Sr be the 0-cube in Sr ∩ N (HN′ ) that is separated from γ ′ (r) by
each Hi′ with N ≤ i ≤ r, as shown in Figure 7–3. Then there is a path in the
image of Dr joining c to xo that crosses only N of the dual curves mapping to
hyperplanes Hi′ : simply travel along the carrier of the dual curve KN mapping
to HN , until arriving at γ ′ (N − 1). Then travel along γ ′ to xo . Hence every
dual curve in Dr separating c from xo maps to a hyperplane that crosses H0 .
Thus ur ≥ r − R − N for all r ≥ 0, and thus |Sr | − dX (γ(r), γ ′ (r)) is unbounded
as r → ∞, a contradiction. We conclude that η(v, v ′ ) ≤ 1.
Proposition 7.2.7. Suppose that γ, γ ′ are combinatorial geodesic rays
emanating from xo , and suppose there exists N, M < ∞ such that for all
r ≥ 0,
div(γ, γ ′ )(r) ≤ 2N r + M,
i.e. suppose that the pair γ, γ ′ has linear divergence. Then η(v, v ′ ) ≤ N .
Proof. First fix r ≥ 0, and let Sr be a shortest r-avoiding path joining γ(r)
⌈ ⌉
to γ ′ (r), so that |Sr | ≤ 2N r + M . Let k = M
. Let Sr = P1r . . . PNr be a
N
concatenation of combinatorial paths such that each Pir has length at most
2r + k, and ||Pir | − |Pjr || ≤ 1 for all i, j. In other words, subdivide Sr into N
paths, each of length as close as possible to
|Sr |
.
N
For 1 ≤ i ≤ N , let fi (r) be the terminal 0-cube of Pir , let f0 (r) =
γ(r), fN (r) = γ ′ (r). Let Qi (r) be a geodesic segment joining fi (r) to xo , for
0 ≤ i ≤ n, so that Q0 (r), QN (r) are hyperplane-equivalent to the initial
length-r segments of γ, γ ′ respectively.
Let ti (r) be the number of hyperplanes crossing both Qi (r) and Qi+1 (r).
Let pi (r) be the number of hyperplanes dual to at least 2 1-cubes of Pir . Then
2r − ti (r) + 2pi (r) ≤ 2r + k, so that 2ti (r) ≥ 2pi (r) − k.
By Lemma 2.1.1, we may choose combinatorial geodesic rays γi , for
1 ≤ i ≤ N such that γi (r) = fi (r) for arbitrarily large values of r. If ti (r) is
unbounded as r → ∞, then γi , γi+1 have infinitely many common hyperplanes,
and thus the associated simplices vi , vi+1 ⊂ ∂△ X satisfy η(vi , vi+1 ) ≤ 1. If
165
ti (r) is uniformly bounded as r → ∞, then so is pi (r), and thus Pi (r) fails
to be a geodesic segment by a uniformly bounded number of 1-cubes, whence
η(vi , vi+1 ) ≤ 1 by Lemma 7.2.6. Hence there is a simplex path
v, v1 , v2 , . . . , vN = v ′
in ∂△ X, and thus η(v, v ′ ) ≤ N .
7.2.3
Divergence and the simplicial boundary
We emphasize that, in this section, linearity of the divergence of X means
that there exists A ∈ R such that for each pair of combinatorial geodesic rays
γ, γ ′ , we have r−1 div(γ, γ ′ )(r) ≤ A + o(1) for all r ≥ 0. The stronger and more
natural notion of linear divergence is considered in a cocompact situation in
the next section.
Theorem 7.2.8. Let X be a strongly locally finite, fully visible, leafless,
one-ended CAT(0) cube complex. Then the following are equivalent:
1. The divergence of X is linear.
2. (∂△ X)1 has finite diameter.
Proof. Note that the 1-skeleton of ∂△ X has finite diameter if and only if
η(v, v ′ ) is uniformly bounded, and that (1) ⇔ (2) does not require that X be
leafless or one-ended, but does require full visibility.
Let v, v ′ ⊂ ∂△ X be simplices. Since X is fully visible, there exists a pair
γ, γ ′ of geodesic rays with common basepoint, respectively representing v, v ′ .
By Proposition 7.2.3,
div(γ, γ ′ )(r)
≤ η(v, v ′ ) + 1 + o(1) ≤ diam((∂△ X)1 ) + o(1).
2r
Hence (2) ⇒ (1).
Conversely, by full visibility, any pair of simplices v, v ′ is represented by
a pair of combinatorial geodesic rays γ, γ ′ . If γ, γ ′ have divergence at most
2N r + M , then η(v, v ′ ) ≤ N by Proposition 7.2.7. Hence (1) ⇒ (2).
166
The stronger notion of f -divergence says, simply, that X has divergence
at most f : N → R if for all r ≥ 1, xo ∈ X0 , and rays γ, γ ′ emanating from xo ,
div(γ, γ ′ )(r) ≤ f (r).
This is more difficult to achieve, and we therefore restrict to the cocompact
case.
7.3
Application to divergence of cocompactly cubulated groups
In order to discuss the divergence of a group, we must introduce the
following more robust notion of divergence, following the discussion in [BC11].
Definition 7.3.1 (Divergence). Let (M, d) be a geodesic space. Given a linear
function ρ(k) = Ak − B with A ∈ (0, 1), B ≥ 0 and points a, b, c ∈ M with
k = d(c, {a, b}), let
div(a, b, c, ρ) = inf |P |
P
where P varies over all paths in M − Nρ(k) (c) joining a to b. The divergence of
M with respect to ρ is
Div(M, ρ)(r) = sup{div(a, b, c, ρ) : d(a, b) ≤ r}.
c
We say that M has f -divergence for some function f if there exists ρ for
which Div(M, ρ)(r) ≤ f (r) for all r ≥ 0, and otherwise, M has super-f
divergence. If α is a bi-infinite geodesic in M , then the divergence of α is the
function div(α(r), α(−r), α(0), ρ)(r). If M has linear divergence with respect
to some ρ, then each pair of geodesic rays with common basepoint, and each
bi-infinite geodesic, has linear divergence.
Note that, if γ, γ ′ are combinatorial geodesic rays with common initial
point, then
div(γ, γ)(r) = div(γ(r), γ ′ (r), γ(0), ρ)
where ρ(r) = r − 1 and M = X1 .
Using the results of the previous section, we can describe the divergence
of geodesics in the median graph X1 .
167
Corollary 7.3.2. Let X be a strongly locally finite, fully visible CAT(0) cube
complex. Let α → X1 be a bi-infinite combinatorial geodesic. Then α has
linear divergence if and only if the simplices v, v ′ of ∂△ X representing the ends
of α lie in the same component of ∂△ X.
Proof. Let γ, γ ′ be geodesic rays with common basepoint whose union is α.
Let v, v ′ be represented by γ, γ ′ , respectively. By Theorem 7.2.8, v, v ′ lie in the
same component if and only if div(γ, γ ′ )(r) ≤ M r + N for all r ≥ 0 and some
M, N ∈ R.
Hence, if v, v ′ lie in the same component of ∂△ X, then for all A < 1, B ≥
0, and all r ≥ 0,
div(γ, γ ′ , γ(r), Ar − B)(r) ≤ M r + N,
so that α has linear divergence.
Conversely, suppose that there exist 0 < A < 1, B ≥ 0 such that for all
r ≥ 0, the preceding inequality holds. Then γ( B+r
) and γ ′ ( B+r
) are joined
A
A
by an r-avoiding path Pr → X, for each r ≥ 0, with |Pr | ≤ M r + N . Let Qr
) and let Q′r be the subpath of γ ′
be the subpath of γ joining γ(r) to γ( B+r
A
) to γ ′ (r). Then Qr Pr Q′r is an r-avoiding path joining γ(r) to
joining γ ′ ( B+r
A
γ ′ (r), so that
(
)
B
div(γ, γ ′ )(r) ≤ |Qr | + |Q′r | + M r + N = 2(A−1 − 1) + M r + N + 2 ,
A
so that v, v ′ lie in the same component of ∂△ X, by Theorem 7.2.8.
We now turn to cubulated groups. Our main result, Theorem 7.3.6,
describes the divergence, with respect to any word-metric, of a (finitelygenerated) group acting properly and cocompactly on a geodesically complete
CAT(0) cube complex.
Definition 7.3.3 (Combinatorially geodesically complete). X is combinatorially geodesically complete if, for every finite set W1 , . . . , Wn of pairwisecrossing hyperplanes, and every section f : {W1 , . . . , Wn } → {W1 , . . . , Wn }± of
168
∩
π|{Wi } , we have that ni=1 f (Wi ) contains 0-cubes arbitrarily far from any fixed
∩
0-cube c ∈ i N (Wi ).
Lemma 7.3.4. The finite-dimensional, strongly locally finite CAT(0) cube
complex X is combinatorially geodesically complete if and only if, for all
geodesic segments P → X, there exists a bi-infinite geodesic γ containing P .
Proof. Suppose X is combinatorially geodesically complete. Let a, m be the
endpoints of P , and let W(a, m) be the finite set of hyperplanes crossing
P . For each W ∈ W(a, m), let W + be the halfspace containing a. The
∩
intersection W ∈W(a,m) W + is unbounded; apply the definition of geodesic
completeness to pairwise-crossing elements of W(a, m) that are maximal in
the partial-ordering induced by the inclusion of halfspaces W + . Thus there
∩
is a geodesic ray α in W ∈W(a,m) W + emanating from a. Let γ+ = P α.
Then γ+ (|P |) = a, and γ+ (0) = m. Moreover, if H is a hyperplane dual
to two distinct 1-cubes of γ+ , then H crosses P and α. Since it crosses P ,
H ∈ W(a, m). Since α ⊂ H + , H cannot cross α. Thus γ+ is a geodesic ray
∩
extending P . Likewise, by considering W ∈W(a,m) W − , we construct a ray
γ− = P β extending P . If H is a hyperplane crossing α and β, then H crosses
every element of W(a, m). By choosing H to be innermost, we can re-choose α
∩
in W ∈W(a,m) W + ∩ H + so that no hyperplane crosses α and β.
Conversely, let W1 , . . . , Wn be a set of pairwise-crossing hyperplanes.
Choose a section f of π and let f ∗ (Wi ) = X − f (Wi ) for each i ≤ n. Choose
∩
∩
a 0-cube c∗ ∈ i f ∗ (Wi ) and a 0-cube c ∈ i f (Wi ). Let P be a geodesic
segment joining c∗ to c. Let γ be a bi-infinite geodesic containing P . Then
∩
i f (Wi ) contains points of γ arbitrarily far from c.
Lemma 7.3.5. Let G act properly, leaflessly, and cocompactly on the combinatorially geodesically complete CAT(0) cube complex X. Suppose that for
some A, B ∈ R, and all combinatorial geodesic rays γ, γ ′ in X,
div(γ, γ ′ )(r) ≤ Ar + B
169
for all r ≥ 0. Then for some A′ , B ′ ∈ R, we have
Div(G, ρ)(r) ≤ A′ r + B ′
for all r ≥ 0, where ρ is some linear function. In other words, if every pair of
geodesic rays in X diverges uniformly linearly, then G has linear divergence
function.
Proof. Recall the usual notion of equivalence of functions: if f, g : N → R
are two functions, then f dominates g if there exists C ≥ 1 such that
g(r) ≤ Cf (Cn) + Cn + c for all r ≥ 0. We say that f is equivalent to g if g
dominates f and f dominates g. If M, N are quasi-isometric metric spaces,
and Div(M, ρ)(r) is bounded above by f (r) for some ρ, then, for some g,
equivalent to f , there exists a linear function ρ′ such that Div(N, ρ′ )(r) is
bounded by g(r). Thus, since G acts properly and cocompactly on X1 , it is
enough to show that Div(X1 , ρ)(r) is bounded by a linear function of r.
Let a, b, c ∈ X0 , with dX (a, b) ≤ r and dX (c, {a, b}) = k. Fix ρ(k) =
Kk − L, for K ∈ (0, 1), L ≥ 0 and let Nk be the closed ρ(k)-ball in X1
centered at c. Let m = m(a, b, c) be the median of a, b, c. If m ̸∈ Nr , then
the geodesic segment in X from a to b, passing through m, is Nk -avoiding, so
that div(a, b, c, ρ)(r) = dX (a, b) ≤ r. Indeed, if y lies on this segment, then
dX (y, m) + dX (m, c) = dX (y, c) > ρ(k).
Therefore, suppose that m ∈ Nk . Let da = dX (m, a), db = dX (m, b),
so that da + db ≤ r. By geodesic completeness, there exist combinatorial
geodesic rays γa , γb with γa (0) = γb (0) = m and γa (da ) = a, γb (db ) = b. Let
nk = dX (m, c) ≤ ρ(k). Choose t = 2ρ(k) + 1, so that ca = γa (t), cb = γb (t) lie
outside of Nk . Moreover, for all x ∈ Nk , we have dX (x, m) ≤ 2ρ(k).
By hypothesis, there is a path P of length at most A(2ρ(k) + 1) + B
joining ca to cb and lying outside the (2ρ(k) + 1)-ball about m, and hence
outside of Nk . Hence
div(a, b, c, ρ)(k) ≤ A (2ρ(k) + 1) + B + |t − da | + |t − db |.
170
for all a, b, c such that dX (a, b) ≤ r and m = m(a, b, c) ∈ Nk . Now, since
m ∈ Nk , we have
k ≤ dX (a, m) + ρ(k)
k ≤ dX (b, m) + ρ(k)
dX (a, m) + dX (b, m) ≤ r,
so that
k≤
r + 2L
,
2(1 − K)
and thus
ρ(k) ≤
K(r + 2L)
+ L.
2(1 − K)
Combining this with the above bound shows that Div(X1 , ρ)(r) is linear.
Theorem 7.3.6. Let the one-ended group G act properly, leaflessly and
cocompactly on the combinatorially geodesically complete cube complex X.
Then G has linear divergence if and only if diam(ΓX) < ∞. Otherwise, the
divergence of some geodesic in X, and hence of G, is at least quadratic. If G
has linear divergence, then ∂△ X decomposes as a nontrivial join. Otherwise,
∂△ X is disconnected.
Proof. Bounded contact graph implies linear divergence: If ΓX is
bounded, then by Theorem 6.3.6, we have X ∼
= Q1 × Q2 , where each Qi is an
unbounded convex subcomplex of X. Let γ, γ ′ be geodesic rays with common
initial 0-cube. By Lemma 7.2.5,
div(γ, γ ′ )(r) ≤ 8r
for all r ≥ 0. Hence Div(G, ρ) is linear for some linear function ρ, by
Lemma 7.3.5. The product decomposition gives a join decomposition of ∂△ X,
by Proposition 5.1.10.
Unbounded contact graph implies at least quadratic divergence:
We adapt the proof of Corollary 5.4 in [BC11]. Suppose ΓX has infinite
171
diameter. Then by Theorem 6.3.6, there exists a rank-one isometry g ∈ G
with a combinatorial axis α such that diamΓX (W(α)) = ∞. Let xo = α(0),
and let γ, γ ′ be the disjoint sub-rays of α emanating from xo . Let v, v ′ ∈ ∂△ X
be the simplices represented by γ, γ ′ respectively. Then by Theorem 6.3.1, v
and v ′ lie in distinct components of ∂△ X. By Theorem 7.2.8, for any choice of
A, B ∈ R, there exists r ≥ 0 such that
div(γ, γ ′ )(r) ≥ Ar + B,
i.e. α has super-linear divergence. This is true of any axis α of g. Now g has
an axis β : R → X in the locally compact CAT(0) space X which is geodesic
with respect to the CAT(0) metric. By Lemma 7.3.7, α can be chosen so
that, if β has linear divergence, then div(γ, γ)(r) is linear. Since for any
choice of α, this is not the case, β must have super-linear divergence. By
Proposition 3.3 of [KL98], β has at least quadratic divergence, and thus G has
at least quadratic divergence.
Lemma 7.3.7. Let G and X be as in Theorem 7.3.6. Let g ∈ G and let
β : R → X be a CAT(0) geodesic axis for g. Then there exists a combinatorial
geodesic axis α for g such that α fellow-travels with β. If β has linear divergence, then div(α([0, ∞)), α((−∞, 0]))(r) is bounded by a linear function of
r.
Proof. There is a CAT(0) geodesic segment B of length L < ∞ such that,
for all t ∈ R, we uniquely express β(t) by first writing t = nL + t′ , where
n ∈ Z and 0 ≤ t′ < L, and then write β(t) = g n B(t′ ). Let U1 , U2 , . . . , Um be
the hyperplanes crossing B, where Ui ∩ B separates Ui−1 ∩ B from Ui+1 ∩ B
for 2 ≤ i ≤ m − 1. Choose a 0-cube x ∈ N (U1 ) in the same halfspace of
U1 as B(0), and let A be a combinatorial geodesic segment emanating from
x that crosses exactly the set U1 , . . . , Um of hyperplanes and joins x to gx.
∪
Let α = n∈Z g n A. Then every hyperplane crossing α is of the form g n Ui ,
and hence the set of hyperplanes crossing α is precisely the set of hyperplanes
172
crossing β. Thus α is a bi-infinite combinatorial geodesic axis, since β crosses
each hyperplane at most once, and ⟨g⟩ acts on α by translations.
Moreover, for any t ∈ R, we have dX (α(t), β(t)) ≤ K, where K is the
maximum distance between a point A(t) and the corresponding point on B.
Hence α and β fellow-travel.
Suppose that for some A < 1, B ≥ 0, there exist C, D ∈ R such that, for
all r ≥ 0, there is a path Pr in X of (CAT(0)) length at most Cr + D that
joins β(r) to β(−r) and does not travel inside of the (Ar − B)-ball about β(0).
Then there exists a path PC ′ r+D′ , where C ′ , D′ depend only on A, B, C, D and
the quasi-isometry constants of the inclusion (X1 , dX ) ,→ (X, dX ), such that
PC ′ r+D′ joins β(C ′ r + D′ ) to β(−C ′ r − D′ ) and avoids the (r − 1)-ball in X1
about α(0). Hence
div(α((−∞, 0]), α([0, ∞)))(r) ≤ 2K + 2D′ + 2(C ′ − 1)r + C(C ′ r + D′ ) + D,
i.e. α has linear divergence.
Combinatorial geodesic completeness is weaker than CAT(0) geodesic
completeness. If every CAT(0) geodesic segment in X extends to a biinfinite CAT(0) geodesic, then it is not hard to verify that the requirement of
Definition 7.3.3 is satisfied. On the other hand, R ×[− 12 , 12 ]2 is combinatorially
geodesically complete but contains CAT(0) geodesic segments that do not
extend.
Theorem 7.3.6 allows us to recover the following result of BehrstockCharney about divergence of right-angled Artin groups from [BC11].
Corollary 7.3.8. Let Θ be a finite simplicial graph and let A be the associated
right-angled Artin group. Then A has linear divergence if and only if Θ
decomposes as the join of two proper subgraphs. Otherwise, the divergence
of G is at least quadratic, and in fact there is a geodesic in G with at least
quadratic divergence.
173
First note that the Salvetti complex of a right-angled Artin group
is CAT(0) geodesically complete, and thus combinatorially geodesically
complete. Corollary 7.3.8 follows immediately from Theorem 7.3.6 once it
is established that ΓXΘ has finite diameter if and only if Θ is a join. This
follows from Theorem 6.3.6 and from Corollary 5.2 of [BC11]. Alternatively,
∂△ Xθ is a join only if XΘ is a cubical product, by Proposition 5.1.10, which
implies that Θ is a join. Conversely, if Θ is a join, then X is a product,
whence ∂△ XΘ is a join. But ∂△ XΘ is a join if and only if AΘ has linear
divergence, by Theorem 7.3.6. Behrstock and Charney also show, using results
of Behrstock-Druţu [BD11] and Behrstock-Druţu-Mosher [BDM09], that if Θ
is connected, then A has at most quadratic divergence.
174
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