HIERARCHICAL HYPERBOLICITY OF ALL CUBICAL GROUPS
MARK F HAGEN AND TIM SUSSE
Let X be a proper CAT(0) cube complex admitting a proper cocompact action by
a group G. Then X has a factor system in the sense of [BHS14]. By results in [BHS14], it follows
that G is a hierarchically hyperbolic group ; this answers questions raised in [BHS14, BHS15].
Our results also resolve a conjecture of Behrstock-Hagen on boundaries of cube complexes.
Abstract.
Introduction
Considerable work in geometric group theory revolves around generalizations of Gromov hyperbolicity: relatively hyperbolic groups, weakly hyperbolic groups, acylindrically hyperbolic
groups, coarse median spaces, semihyperbolicity, lacunary hyperbolicity, etc.
In recent years
much attention has been paid to groups acting properly and cocompactly on CAT(0) cube
complexes, which are important examples of metric nonpositive curvature and which also have
features reminiscent of hyperbolicity. Such complexes give a combinatorially and geometrically
rich framework to build on, and many groups have been shown to admit such actions (for a
small sample, see [Sag95, Wis04, OW11, BW12, HW15]).
Many results follow from studying the geometry of CAT(0) cube complexes, often using strong
properties reminiscent of negative curvature.
For instance, several authors have studied the
structure of quasiats and Euclidean sectors in cube complexes, with applications to rigidity
properties of right-angled Artin group [Xie05, BKS08, Hua14].
These spaces have also been
shown to be median [Che00] and to have only semi-simple isometries [Hag07]. Further, under
reasonable assumptions, a CAT(0) cube complex
Isom(X )
X
either splits as a nontrivial product or
must contain a rank-one element [CS11].
Once a given group is known to act properly and cocompactly on a CAT(0) cube complex
the geometry of the cube complex controls the geometry and algebra of the group. For instance,
such a group is biautomatic and cannot have Khazdan's property (T) [NR98, NR97], and it
must satisfy the Tits alternative [SW05]
Here, we examine the geometry of cube complexes admitting proper, cocompact group actions
from the point of view of their convex subcomplexes, showing that such a complex has strong
geometric restrictions. These translate into properties of the group via our main application:
Corollary A. Let G act properly and cocompactly on the proper CAT(0) cube complex X . Then
X is a hierarchically hyperbolic space and G is a hierarchically hyperbolic group.
Hierarchically hyperbolic spaces/groups
(HHS/G's), introduced in [BHS14, BHS15]. HHS/G's
were proposed as a common framework for studying mapping class groups and (certain) cubical
groups. Knowledge that a group is hierarchically hyperbolic has strong consequences, detailed
below, that imply many of the nice properties of mapping class groups. Our results place all
cubical groups in this framework as well, immediately implying many of these properties, and
lling in a major gap in the theory of hierarchically hyperbolic spaces.
Date : September 5, 2016.
Hagen was supported by the Engineering and Physical Sciences Research Council grant of Henry Wilton.
Susse was partially supported by National Science Foundation grant DMS-1313559.
1
HIERARCHICAL HYPERBOLICITY OF ALL CUBICAL GROUPS
2
A main goal of [BHS14] was to develop a theory of the coarse geometry of CAT(0) cube
complexes analogous to the Masur-Minsky theory for mapping class groups [MM99, MM00].
This goal was achieved in the context of cube complexes
X
with
factor systems.
A
factor
system S on X is a uniformly nite-multiplicity collection of convex subcomplexes that contains
X , contains each combinatorial hyperplane, and contains gF (F 0 ) whenever F, F 0 ∈ S, where
gF : X → F is combinatorial closest-point projection. When they exist, factor systems provide
the necessary analogue of the collection of all subsurfaces of a surface.
The reader will observe that it is the uniformly nite-multiplicity condition that can fail to
0cube on the bottom step B is conwhere R is a horizontal combinatorial
point along B . Our main theorem rules
hold: the simplest example is the staircase! Indeed, any
tained in a collection of subcomplexes of the form
gB (R),
hyperplane, whose cardinality grows as one moves the
this out for CAT(0) cube complexes with proper, cocompact group actions:
Theorem B.
Let X be a proper CAT(0) cube complex and let G ≤ Aut(X ) act properly and
cocompactly on X . Then X contains a factor system.
Theorem B and results of [BHS14] (see Remark 13.2 of that paper) together imply Corollary A,
positively answering Question 8.12 of [BHS14] and part of Question A of [BHS15]. Previously,
many CAT(0) cube complexes were known to have factor systems and thus be hierarchically
hyperbolic, but it was not known that a geometric action guarantees a factor system.
One of the key technical insights of this paper is the following.
complex, the
First, in a CAT(0) cube
hyperclosure F is the set of convex subcomplexes one should consider when trying to
nd a factor system, namely the smallest set of subcomplexes that contains each combinatorial
hyperplane and contains
gF (F 0 )
whenever it contains
F, F 0 .
The hyperclosure
F
is shown in
Section 2 to have a graded structure: the lowest-grade elements are combinatorial hyperplanes,
then we add projections of combinatorial hyperplanes to combinatorial hyperplanes, etc. This
allows for several arguments to proceed by induction on the grade. Essentially by denition,
orthogonal complement of a 1cube e, i.e. a maximal
e × H as a subcomplex. We show, in
that, more generally, F is precisely the set of convex subcomplexes F such that
compact, convex subcomplex C so that the orthogonal complement of C is F .
a combinatorial hyperplane
convex subcomplex
Theorem 3.5,
there exists a
H
H
is the
for which
Applications of Corollary A.
X
contains the product
Corollary A has numerous consequences for a group
geometrically on a CAT(0) cube complex
X.
Denote by
S
G
acting
the factor system provided by
F ∈ S is associated to a graph CF which
factored contact graph ). The import of
Corollary A is that X , equipped with this set of quasi-trees, and coarse projections πF : X → CF
Theorem B and recall from [BHS14] that each
is uniformly quasi-isometric to a simplicial tree (the
explained in [BHS14], satises a list of axioms reminiscent of theorems about mapping class
groups and curve graphs. Results about
G, X
which previously required one to hypothesize a
factor system but which, in view of Corollary A, do not really need that hypothesis, include:
•
In combination with [BHS14, Corollary 14.5], Corollary A shows that
cally on the contact graph of
X,
G
acts acylindri-
i.e. the intersection graph of the hyperplane carriers,
which is a quasi-tree [Hag14].
•
Corollary A combines with Theorem 9.1 of [BHS14] to provide a Masur-Minsky style
G:
X
x to gx, where
g ∈ G, is given by summing the distances between the projections of x, gx to a collection
distance estimate in
up to quasi-isometry, the distance in
from
of uniform quasi-trees associated to the elements of the factor system.
•
Corollary A combines with Proposition 11.4 of [BHS14] to produce a
quasi-isometric embedding of
X
Gequivariant
into the product of nitely many quasi-trees, under a
mild hypothesis on the group action.
HIERARCHICAL HYPERBOLICITY OF ALL CUBICAL GROUPS
•
3
Corollary A combines with Corollary 9.24 of [DHS16] to prove that either
a convex subcomplex of
X
G
stabilizes
splitting as the product of unbounded subcomplexes, or
contains an element acting loxodromically on the contact graph of
X.
G
This is a new
proof of the cocompact version of the Caprace-Sageev rank-rigidity theorem [CS11].
We now turn to applications of Theorem B that do not involve hierarchical hyperbolicity.
Applications of Theorem B to boundaries of
structure of the boundary of
X.
X.
Theorem B also gives insight into the
To motivate these applications, we rst mention an aggravating
geometric/combinatorial question about cube complexes which is answered by our results.
Z dened as follows. First, a ray-strip is a square
Sn = [n, ∞) × [− 12 , 12 ], with the product cell-structure where [n, ∞) has
0skeleton {m ∈ Z : m ≥ n} and [− 12 , 12 ] is a 1cube. To build Z , choose an increasing sequence
(an )n of integers, collect the ray-strips San ∼
= [an , ∞) × [− 21 , 12 ], and identify [an+1 , ∞) × {− 12 } ⊂
1
San +1 with [an , ∞) × { 2 } ⊂ San for each n. The model staircase is the cubical neighborhood of
2
a Euclidean sector in the standard tiling of E by squares, with one bounding ray in the xaxis,
but for certain (an )n , Z may not contain a nontrivial Euclidean sector. From our combinatorial
point of view, this makes no dierence. (One can dene a d-dimensional staircase by choosing
(1)
(d−1)
d − 1 sequences of integers (an )n , . . . , (an
)n and considering the constructions above, gluing
(1)
(d−1)
together steps of the form [ai , ∞) × · · · × [ai
, ∞) × [− 12 , 12 ].) We will see below that the set
of horizontal hyperplanes in Z see Figure 1 for the meaning of horizontal is interesting
because there is no geodesic ray in Z crossing exactly the set of horizontal hyperplanes.
A
staircase
is a CAT(0) cube complex
complex of the form
Figure 1. Part of a staircase.
Now let
X
be a proper CAT(0) cube complex with a group
Can there be a convex staircase subcomplex in
X?
G acting properly and cocompactly.
A positive answer seems very implausible,
but, surprisingly, this question was open. Corollary C, a consequence of our main result, implies:
Corollary.
Let X be a proper CAT(0) cube complex admitting a proper cocompact group action.
Then X does not contain a convex staircase.
The more general version of this statement involves the boundary of X . Specically, the
simplicial boundary ∂4 X of a CAT(0) cube complex X was dened in [Hag13]. The simplices
of ∂4 X correspond to equivalence classes of innite sets H of hyperplanes such that:
• if H, H 0 ∈ H are separated by a hyperplane V , then V ∈ H;
• if H1 , H2 , H3 ∈ H are disjoint, then one of H1 , H2 , H3 separates the other two;
• for H ∈ H, at most one halfspace associated to H contains innitely many V ∈ H.
These boundary sets are partially ordered by coarse inclusion (i.e., A B if all but nitely
many hyperplanes of A are contained in B ), and two are equivalent if they have nite symmetric
dierence; ∂4 X is the simplicial realization of this partial order. The motivating example of
a simplex of ∂4 X is: given a geodesic ray γ of X , the set of hyperplanes crossing γ has the
preceding properties. Not all simplices are realized by a geodesic ray in this way: a simplex in X
is called visible if it is represented by the collection of hyperplanes crossing some combinatorial
geodesic ray. For example, if Z is a staircase, then ∂4 Z has an invisible 0simplex, represented
by the set of horizontal hyperplanes. However, using Theorem B, we resolve Conjecture 2.8 of
[BH16], ruling out invisible simplices in the presence of geometric group actions:
HIERARCHICAL HYPERBOLICITY OF ALL CUBICAL GROUPS
4
Corollary C. Let X be a proper CAT(0) cube complex which admits a proper and cocompact
group action. Then every simplex is ∂4 X is visible, that is, it is equivalent to the set of hyperplanes crossing an innite geodesic ray. Moreover, let v ∈ ∂4 X be a 0simplex. Then there
exists a CAT(0) geodesic ray γ such that the set of hyperplanes crossing γ represents v .
The moreover part follows from the rst part and [Hag13, Lemma 3.32]. Corollary C combines with [BH16, Theorem 5.13] to imply that
G,
divergence for
∂4 X
detects thickness of order 1 and quadratic
without the visibility hypothesis imposed in [BH16].
plies the corollary about staircases at the beginning of this paper: if
X,
then the horizontal hyperplanes represent an invisible
Z
0simplex
Corollary C also im-
is a convex staircase in
of
∂4 X ,
a contradiction.
More generally, we obtain the following from Corollary C and a simple argument in [Hag13]:
Corollary D.
Let γ be a CAT(0)-metric or combinatorial geodesic ray in X , where X is as in
Corollary C and the set of hyperplanes crossing γ represents a d-dimensional simplex of ∂4 X .
Then there exists a combinatorially isometrically embedded d+1-dimensional orthant subcomplex
O ⊆ Hull(γ). Moreover, γ lies in a nite neighborhood of O.
(A k -dimensional orthant subcomplex is a CAT(0) cube complex isomorphic to the product of
k copies of the standard tiling of [0, ∞) by 1cubes, and the convex hull Hull(A) of a subspace
A ⊆ X is the smallest convex subcomplex containing A.)
Corollary D is related to Lemma 4.9 of [Hua14] and to statements in [Xie05, BKS08] about
Euclidean sectors in cocompact CAT(0) cube complexes and arcs in the Tits boundary.
In
particular it shows that in any CAT(0) cube complex with a proper cocompact group action,
nontrivial geodesic arcs on the Tits boundary can be extended to a arcs of length at least
Questions.
Theorem B suggests questions. The rst asks whether Theorem B is sharp:
Question 1.
•
If
X
π/2.
Let
X
be a CAT(0) cube complex and let
is locally nite and
G
G ≤ Isom(X ).
acts cocompactly (not necessarily properly), must
X
contain
a factor system?
•
If
X
is uniformly locally nite and
must
X
G
acts essentially with no xed point in
X ∪ ∂X ,
contain a factor system?
Modifying the hypotheses in these ways is reasonable since cocompactness is used in one
isolated spot in the proof of Theorem B, namely Lemma 2.8.
Positive answers to the above
questions would allow one to carry out the HHS proof of the rank-rigidity theorem from [DHS16]
at the level of generality of [CS11]. Finally, in [BHS15], it was suggested that Theorem B actually
follows from a stronger statement. Specically:
Question 2.
action. Does
Let
X
X
be a CAT(0) cube complex which admits a proper and cocompact group
embed a convex subcomplex of the universal cover of the Salvetti complex of
some right-angled Artin group?
We wonder whether the techniques of the present paper could be used to attack the above
question, which is of considerable independent interest.
Plan of the paper.
Section 1 contains background on CAT(0) cube complexes closely follow-
ing [BHS14]. Section 2 studies the
hyperclosure,
a collection of convex subcomplexes, denable
in any CAT(0) cube complex, which we will show has bounded multiplicity in order to prove
Theorem B. In Section 3, we relate subcomplexes in the hyperclosure to the orthogonal complement operation on convex subcomplexes. Section 4 combines these ingredients into a proof of
Theorem B, and in Section 5 we complete the proof of Corollary C and Corollary D.
HIERARCHICAL HYPERBOLICITY OF ALL CUBICAL GROUPS
Acknowledgements.
5
MFH thanks: Jason Behrstock and Alessandro Sisto for discussions of
factor systems during our work on [BHS14]; Nir Lazarovich and Dani Wise for discussions on
Question 2, which remains open; Dan Guralnik, Alessandra Iozzi, Yulan Qing, and Michah
Sageev for discussions about staircases. We thank Richard Webb and Henry Wilton for helping
to organize the conference
Beyond Hyperbolicity
(Cambridge, June 2016), at which much of the
work on this paper was done. Both of the authors thank Franklin for his sleepy vigilance.
1. Background
1.1.
Basics on CAT(0) cube complexes.
Recall that a
CAT(0) cube complex X
is a simply-
connected cube complex in which the link of every vertex is a simplicial ag complex (see
e.g. [BH99, Chapter II.5], [Sag14, Wis, Che00] for precise denitions and background). In this
paper,
X
always denotes a CAT(0) cube complex.
Our choices of language and notation for
describing convexity, hyperplanes, gates, etc. follow the account given in [BHS14, Section 2].
Denition 1.1 (Hyperplane, carrier, combinatorial hyperplane). A midcube in the unit cube
c = [− 12 , 12 ]n is a subspace obtained by restricting exactly one coordinate to 0. A hyperplane in
X is a connected subspace H with the property that, for all cubes c of X , either H ∩ c = ∅ or
H ∩c consists of a single midcube of c. The carrier N (H) of the hyperplane H is the union of all
closed cubes c of X with H ∩c 6= ∅. The inclusion H → X extends to a combinatorial embedding
∼
=
H × [− 21 , 21 ] −→ N (H) ,→ X identifying H × {0} with H . Now, H is isomorphic to a CAT(0)
± of
cube complex whose cubes are the midcubes of the cubes in N (H). The subcomplexes H
1
N (H) which are the images of H ×{± 2 } under the above map are isomorphic as cube complexes
to H , and are combinatorial hyperplanes in X . Thus each hyperplane of X is associated to two
combinatorial hyperplanes lying in N (H).
Remark.
The distinction between hyperplanes (which are not subcomplexes) and combina-
A ⊂ X , either a convex subcomplex or a
H crosses A to mean that H ∩ A 6= ∅.
hyperplane H is precisely the set of hyperplanes
torial hyperplanes (which are) is important. Given
H,
hyperplane, and a hyperplane
we sometimes say
Observe that the set of hyperplanes crossing a
crossing the associated combinatorial hyperplanes.
Denition 1.2 (Convex subcomplex).
cube
c
of
X
whose
0skeleton
A subcomplex
Y⊆X
Y satises c ⊆ Y
X (1) .
lies in
path-metric, is metrically convex in
There are various characterizations of cubical convexity.
CAT(0)metric convexity
for subcomplexes
Denition 1.3 (Convex Hull).
convex if Y is full i.e. every
Y (1) , endowed with the obvious
is
and
Cubical convexity coincides with
[Hag07], but not for arbitrary subspaces.
Given a subspace
A ⊂ X , we denote by Hull(A) its convex hull,
A.
i.e. the intersection of all convex subcomplexes containing
If
Y ⊆ X
is a convex subcomplex, then
have the form
H ∩ Y,
where
H
Y
is a CAT(0) cube complex whose hyperplanes
is a hyperplane of
X,
and two hyperplanes
0
intersect if and only if H, H intersect. Recall from [Che00] that the graph
dX
H ∩ Y, H 0 ∩ Y
X (1) , endowed with
1, is a median graph (and in fact being
1skeleta of CAT(0) cube complexes among graphs): given 0
cubes x, y, z , there exists a unique 0cube m = m(x, y, z), called the median of x, y, z , so that
Hull(x, y) ∩ Hull(y, z) ∩ Hull(x, z) = {m}.
Let Y ⊆ X be a convex subcomplex. Given a 0cube x ∈ X , there is a unique 0cube y
0
so that dX (x, y) is minimal among all 0cubes in Y . Indeed, if y ∈ Y , then the median m of
0
0
x, y, y lies in Y , by convexity of Y , but dX (x, y ) = dX (x, m) + dX (m, y 0 ), and the same is true
0
(0) , we have m = y = y 0 .
for y . Thus, if dX (x, y ) and dX (x, y) realize the distance from x to Y
the obvious path metric
in which edges have length
a median graph characterizes
HIERARCHICAL HYPERBOLICITY OF ALL CUBICAL GROUPS
Denition 1.4
gY : X → Y
.
(Gate map)
For a convex subcomplex
Y ⊆ X,
the
gate map
6
to
Y
is the map
so that:
v ∈ X (0) , gY (v) is the unique 0cube of Y lying closest to v ;
(0) , the map g collapses c to
for any dcube, labeled c, of X with vertices x0 , . . . , x2d ∈ X
Y
0
the unique k cube c in Y with 0cells gY (x0 ) . . . , gY (x2d ) in the natural way, respecting
(1) for all
(2)
the cubical structure.
k in the denition above is the number of hyperplanes that intersect both c
Y . The hyperplanes that intersect c0 are precisely the hyperplanes which intersect both
c and Y . (Indeed, the Helly property ensures that there are cubes crossing exactly this set of
hyperplanes, while convexity of Y shows that at least one such cube lies in Y ; the requirement
0
0
that it contain gY (xi ) then uniquely determines c .) Thus, for any convex subcomplex Y, Y ⊆ X ,
0
0
the hyperplanes crossing gY (Y ) are precisely the hyperplanes which cross both Y and Y . The
The integer
and
next denition formalizes this relationship.
Denition 1.5
.
parallel, written F k F 0 , if
0
for each hyperplane H of X , we have H ∩ F 6= ∅ if and only if H ∩ F 6= ∅. The subcomplex
F is parallel into F 0 if F is parallel to a subcomplex of F 0 , i.e. every hyperplane intersecting F
0
0
intersects F . We denote this by F ,→k F . Any two 0cubes are parallel subcomplexes.
(Parallel)
The convex subcomplexes
F
and
F0
are
The following is proved in [BHS14, Section 2] and illustrated in Figure 2:
Lemma 1.6.
Let F, F 0 be parallel subcomplexes of the CAT(0) cube complex X . Then Hull(F ∪
∼
= F × A, where A is the convex hull of a shortest combinatorial geodesic with endpoints on
F and F 0 . The hyperplanes intersecting A are those separating F, F 0 . Moreover, if D, E ⊂ X
are convex subcomplexes, then gE (D) ⊂ E is parallel to gD (E) ⊂ D.
F 0)
gD (E)
gE (D)
H
D
Figure 2. Here,
D, E
E
V
are convex subcomplexes. The gates
gD (E), gE (D)
are
parallel, and are joined by a product region, shown as a cylinder. Each hyper-
Hull gD (E) ∪ gE (D) either separates gD (E), gE (D) (e.g.
V ) or crosses both of gD (E), gE (D) (e.g. the hyperplane H ).
plane crossing
hyperplane
the
The next Lemma will be useful in Section 2.
Lemma 1.7.
Proof.
For convex subcomplexes C, D, E , we have ggC (D) (E) k gC (gD (E)) k gC (gE (D)).
F = gC (D). Let H be a hyperplane so that H ∩ gF (E) 6= ∅. Then H ∩ E, H ∩ F 6= ∅
H ∩ C, H ∩ D 6= ∅, by the comment following Denition 1.4. Thus gF (E) is parallel
into gC (gD (E)) and gC (gE (D)). However, the hyperplanes crossing either of these are precisely
the hyperplanes crossing all of C, D, E . Thus, they cross F and D , and thus cross gF (E).
Let
and thus
Denition 1.8 (Orthogonal complement).
A ⊆ X be a convex subcomplex. Let PA be the
A, so that PA ∼
= A × A⊥ , where A⊥ is a CAT(0)
(0) , the
cube complex that we call the abstract orthogonal complement of A in X . For any a ∈ A
⊥
convex subcomplex {a} × A is the orthogonal complement of A at a. See Figures 3 and 4.
Let
convex hull of the union of all parallel copies of
HIERARCHICAL HYPERBOLICITY OF ALL CUBICAL GROUPS
7
e⊥
e
Figure 3. Combinatorial hyperplanes are orthogonal complements of
e⊥
1
s⊥
e⊥
2
⊥
(e1 ∪ s ∪ e2 )
e2
e1
s
Figure 4. Orthogonal complements of
Note that
1cubes.
(e1 ∪ e2 ∪ s)⊥ k ge⊥ (ge⊥ (s⊥ ));
2
1cubes e1 , e2
and
2cube s
are shown.
this illustrates Theorem 3.5 below.
1
Lemma 1.9.
Let A ⊆ X be a convex subcomplex. For any a ∈ A, a hyperplane H intersects
{a} × A⊥ if and only if H is disjoint from every parallel copy of A but intersects each hyperplane
V with V ∩ A 6= ∅. Hence {a} × A⊥ , {b} × A⊥ are parallel for all a, b ∈ A(0) .
Proof.
This follows from the denition of
two classes, those intersecting
A
its parallel copies). By denition,
separated from
a
PA :
the hyperplanes crossing
PA
are partitioned into
(and its parallel copies) and those disjoint from
A⊥
is the convex hull of the set of
0cubes
A
of
(and any of
PA
that are
only by hyperplanes of the latter type. The product structure ensures that
any hyperplane of the rst type crosses every hyperplane of the second type.
Hyperclosure and factor systems.
Denition 1.10 (Factor system, hyperclosure).
1.2.
The hyperclosure of X is the intersection F of
F0 of convex subcomplexes of X that satisfy the following three properties:
0
0
(1) X ∈ F , and for all combinatorial hyperplanes H of X , we have H ∈ F ;
0
0
0
0
(2) if F, F ∈ F , then gF (F ) ∈ F ;
0
0
0
0
(3) if F ∈ F and F is parallel to F , then F ∈ F .
Note that F is Aut(X )invariant. If there exists ξ such that for all x ∈ X , there are at most ξ
elements F ∈ F with x ∈ F , then, following [BHS14], we call F a factor system for X .
all sets
Remark 1.11.
The denition of a factor system in [BHS14] is more general than the denition
given above. The assertion that
X
has a factor system in the sense of [BHS14] is equivalent to
the assertion that the hyperclosure of
X
has nite multiplicity, because any factor system (in
the sense of [BHS14]) contains all elements of
F
whose diameter exceed a given xed threshold.
2. Analysis of the hyperclosure
Fix a proper
2.1.
X
with a group
Decomposition.
Let
G
acting properly and cocompactly. Let
F0 = {X }
F1
is the set of
be the hyperclosure.
n ≥ 1, let Fn be the subset of F consisting
gH (F ), where F ∈ Fn−1 and H is a
combinatorial hyperplanes in X
and, for each
of those subcomplexes that can be written in the form
combinatorial hyperplane. Hence
F
HIERARCHICAL HYPERBOLICITY OF ALL CUBICAL GROUPS
Lemma 2.1 (Decomposing F).
8
Each F ∈ F − {X } is parallel to a subcomplex of the form
gH1 (gH2 (· · · gHn−1 (Hn ) · · · ))
for some n ≥ 1, where each Hi is a combinatorial hyperplane, i.e. F/k = (∪n≥1 Fn ) /k .
Proof.
This follows by induction, Lemma 1.7, and the denition of
Corollary 2.2.
Proof.
F.
F = ∪n≥0 Fn .
F ⊆ ∪n≥0 Fn . Let F ∈ F. If F = X , then F ∈ F0 . Otherwise,
by Lemma 2.1, there exists n ≥ 1, a combinatorial hyperplane H , and a convex subcomplex
F 0 ∈ ∪k≤n Fk with F k gH (F 0 ). Consider PF ∼
= F × F ⊥ and choose f ∈ F ⊥ so that F × {f }
0
0
coincides with F . Then F × {f } lies in some combinatorial hyperplane H either H = H and
F = gH (F ), or F is non-unique in its parallelism class, so lies in a combinatorial hyperplane in
⊥
0
0
the carrier of a hyperplane crossing F . Consider gH 0 (gH (F )). On one hand, gH 0 (gH (F )) ∈
0
∪k≤n+1 Fk . On the other hand, gH 0 (gH (F )) = F . Hence F ∈ ∪n≥1 Fn , as required.
2.2.
It suces to show
Stabilizers act cocompactly.
cocompactly on
F
for each
F ∈ F.
The goal of this subsection is to prove that
StabG (F ) acts
The following lemma is standard but we include a proof in
the interest of a self-contained exposition.
Lemma 2.3 (Coboundedness from nite multiplicity). Let X be a metric space and let G →
Isom(X) act cocompactly, and let Y be a Ginvariant collection of subspaces such that every ball
intersects nitely many elements of Y . Then StabG (P ) acts coboundedly on P for every P ∈ Y .
Proof. Let P ∈ Y , choose a basepoint r ∈ X , and use cocompactness to choose t < ∞ so that
d(x, G · r) ≤ t for all x ∈ X . Choose g1 , . . . , gs ∈ G so that the Gtranslates of P intersecting
N10t (r) are exactly gi P for i ≤ s. Since Y is Ginvariant and locally nite, s < ∞. Let
Kr = maxi≤s d(r, gi r). For each g ∈ G, the translates of P that lie within distance 10t of g · r
are precisely gg1 P, . . . , ggs P and Kgr = Kr since d(r, gi · r) = d(g · r, ggi · r).
Fix a basepoint p ∈ P and let q ∈ P be an arbitrary point; choose hp , hq ∈ G so that
d(hp · r, p) ≤ t, d(hq · r, q) ≤ t. Without loss of generality, we may assume that hq = 1. Then
{hp gi P }si=1 is the set of P translates intersecting N10t (hp · r). Now, p ∈ P and d(hp · r, p) < 10t,
so there exists i so that hp gi P = P , i.e. hp gi ∈ StabG (P ). Finally,
d(hp gi · q, p) ≤ d(hp gi · r, hp gi · q) + d(hp · r, p) + d(hp gi · r, hp · r) ≤ 2t + Kr ,
which is uniformly bounded. Hence the action of
StabG (P )
on
P
is cobounded.
Remark 2.4. We use Lemma 2.3 when X and P are proper, to get a cocompact action.
Lemma 2.5. Let X be a proper CAT(0) cube complex with a group G acting cocompactly.
Let
Y, Y 0 ⊂ X be parallel convex subcomplexes, then StabG (Y ) and StabG (Y 0 ) are commensurable.
Thus, if StabG (Y ) acts cocompactly on Y , then StabG (Y ) ∩ StabG (Y 0 ) acts cocompactly on Y 0 .
Proof. Let T be the set of StabG (Y )translates of Y 0 . Then each gY 0 ∈ T is parallel to Y , and
dX (gY 0 , Y ) = dX (Y 0 , Y ). Since Y ⊥ is locally nite, |T | < ∞. Hence K = ker(StabG (Y ) →
Sym(T )) has nite index in StabG (Y ) but lies in StabG (Y ) ∩ StabG (Y 0 ). By Lemma 1.6, K acts
0
0
0
cocompactly on Hull(Y ∪ Y ), stabilizing Y , and thus acts cocompactly on Y .
Denition 2.6. Let H ∈ F1 . For n ≥ 1, k ≥ 0, let Fn,H,k be the set of F ∈ Fn so that F =
gH (F 0 ) for some F 0 ∈ Fn−1 with d(H, F 0 ) ≤ k . Let Fn,H = ∪k≥0 Fn,H,k and Fn,k = ∪H∈F1 Fn,H,k .
Proposition 2.7
(Cocompactness). Let n ≥ 1. Then, for any F ∈ Fn , StabG (F ) acts cocompactly on F . Hence StabG (F ) acts cocompactly on F for each F ∈ F.
HIERARCHICAL HYPERBOLICITY OF ALL CUBICAL GROUPS
Proof.
The second assertion follows from the rst and Corollary 2.2.
n, k
k
9
We argue by double
Fn ,
Ginvariant for all n, k . Similarly, Fn,H,k is StabG (H)invariant for all H ∈ F1 .
Base Case: n = 1. From local niteness of X , cocompactness of the action of G and
Lemma 2.3, we see that StabG (H) acts cocompactly on H for each H ∈ F1 .
Inductive Step 1: (n, k) for all k implies (n + 1, 0). Let F ∈ Fn+1,0 . Then F = H ∩ F 0 ,
0
0
00
00
0
where H ∈ F1 and F ∈ Fn . By denition, F = gH 0 (F ) for some F ∈ Fn−1 and H ∈ F1 .
0
0
Thus K = StabG (F ) acts cocompactly on F by induction.
0
0
Let S = {k(H ∩ F ) : k ∈ K}, which is a K invariant set of convex subcomplexes of F .
Moreover, since the set of all K translates of H is a locally nite collection, because X is locally
0
nite and H is a combinatorial hyperplane, S has the property that every ball in F intersects
0
nitely many elements of S . Lemma 2.3, applied to the cocompact action of K on F , shows
0
that StabK (H ∩ F ) (which equals StabK (F )), and hence StabG (F ), acts cocompactly on F .
Inductive Step 2: (n, k) implies (n, k + 1). Let F ∈ Fn,k+1 so that F = gH (F 0 ) with
H ∈ F1 , F 0 ∈ Fn−1 and d = d(H, F 0 ) ≤ k + 1. If d ≤ k , induction applies. Thus, we can
assume that d = k + 1. Then there is a product region F × [0, d] ⊂ X with F × {0} = F ,
0
0
and F × {d} ⊂ F . Then F1 := F × {1} is a parallel copy of F , and F1 ⊂ gH 0 (F ) with
0
0
0
0
d(H , F ) = d − 1 ≤ k . By induction L = StabG (gH 0 (F )) acts cocompactly on gH 0 (F ).
0
We claim that F1 = gH 0 (F ) ∩ gH 0 (H). To see this, note that the hyperplanes that cross F1
0
are exactly the hyperplanes that cross F and H . However, those are the hyperplanes which
0
0
cross H and F which also cross H . It easily follows that the two subcomplexes are equal.
0
0
Now let T be the set of Ltranslates of F1 = gH 0 (F ) ∩ gH 0 (H) in gH 0 (F ). This is an L
0
0
invariant collection of convex subcomplexes of gH 0 (F ). Moreover, each ball in gH 0 (F ) intersects
nitely many elements of T . Indeed, T is a collection of subcomplexes of the form T` =
g`H 0 (`H) ∩ gH 0 (F 0 ), where ` ∈ L. Recall that dX (H, H 0 ) = 1. Hence, xing y ∈ gH 0 (F 0 )
and t ≥ 0, if {T`i }i∈I ⊆ T is a collection of elements of T , all of which intersect Nt (y), then
{`i H, `i H 0 }i∈I all intersect Nt+1 (y). However, by local niteness of X there are only nitely
0
0
0
many distinct `i H, `i H . Further, if `i H = `j H and `i H = `j H , then T`i = T`j . Thus, the
index set I must be nite. Hence, by Lemma 2.3 and cocompactness of the action of L on
gH 0 (F 0 ), we see (as in Inductive Step 1) that StabG (F1 ) acts cocompactly on F1 . Now, since F1
is parallel to F , we see by Lemma 2.5 that StabG (F ) acts cocompactly on F .
induction on
Fn,k
to prove the rst assertion, with
as in Denition 2.6. First, observe that
are
Ascending chain condition. We reduce Theorem B to a claim about (chains in F.
T
Lemma 2.8 (Finding ascending chains). Let U ⊆ F be an innite subset satisfying U ∈U U 3 x
2.3.
for some x ∈ X . Then there exists a sequence {Fi }i≥1 in F so that x ∈ Fi ( Fi+1 for all i.
Proof. Let Fx ⊇ U be the set of F ∈ F with x ∈ F . Let Ω be the directed graph with vertex set
Fx , with (F, F 0 ) a directed edge if F ( F 0 and there does not exist F 00 ∈ Fx with F ( F 00 ( F 0 .
0
We may assume that there is a unique F0 ∈ Fx with no incoming Ωedges. Indeed, if F0 , F0
have this property, then their intersection lies in Fx . Hence either there is a nite, directed path
0
from F0 ∩ F0 to F0 , contradicting minimality of F0 , or there is an innite chain in Fx consisting
0
of elements that all contain F0 ∩ F0 and are all contained in F0 ; in the latter case, we are done.
Similarly, we can assume that every F ∈ Fx is at nite distance in Ω from F0 .
Let F ∈ Fx and suppose that {Fi }i is the set of vertices of Ω so that (F, Fi ) is an edge. For
i 6= j , we have F ⊆ Fi ∩ Fj ( Fi , so since Fi ∩ Fj = gFi (Fj ) ∈ F, we have Fi ∩ Fj = F .
The set {Fi }i is invariant under the action of StabG (F ). Also, by Proposition 2.7, StabG (F )
acts cocompactly on F . A 0cube y ∈ F is diplomatic if there exists i so that y is joined to a
vertex of Fi − F by a 1cube in Fi ; see Figure 5. Only uniformly nitely many Fi can witness
the diplomacy of y since X is uniformly locally nite and Fi ∩ Fj = F whenever i 6= j . Also, y is
diplomatic, witnessed by Fi1 , . . . , Fik , if and only if gy is diplomatic, witnessed by gFi1 , . . . , gFik ,
for each g ∈ StabG (F ). Since StabG (F ) y F cocompactly, we thus get |{Fi }i / StabG (F )| < ∞.
HIERARCHICAL HYPERBOLICITY OF ALL CUBICAL GROUPS
10
y
F
Fj
Fi
Figure 5. The
0cube y
is diplomatic: incident (bold)
1cubes
reach out into
Fi , Fj .
F ∈ F containing a point of G · x and a directed
b is a graded directed graph as above. For each
Ω
b at distance n from a minimal element. The above
n ≥ 0, let Sn be the set of vertices in Ω
b
argument shows that G acts conitely on each Sn , and thus Ω/G
is locally nite. Hence, by
(0)
b
b
König's innity lemma, either Ω/G contains a directed ray or Ω
/G is nite. In the former
case, Ω must contain a directed ray, in which case there exists {Fi } ⊆ F with Fi ( Fi+1 for all
i. Up to translating by an appropriate element of G, we can assume that x ∈ F1 . The latter
case means that the set of F ∈ F such that F ∩ G · x 6= ∅ is Gnite. But since G acts properly
and cocompactly on X , any Ginvariant Gnite collection of subcomplexes whose stabilizers
act cocompactly has nite multiplicity, a contradiction.
Let
b
Ω
be the graph with a vertex for each
edge for minimal containment as above. Then
Lemma 2.8 shows that Theorem B holds if all ascending chains in
F
are nite.
3. Orthogonal complements
In this section, we rst prove three general lemmas about orthogonal complements of convex
subcomplexes of a CAT(0) cube complex
Lemma 3.1. Let A ⊆ B ⊆ X
Proof.
Let
copies of
B
x ∈ {a} × B ⊥ .
X.
be convex subcomplexes and let a ∈ A. Then {a}×B ⊥ ⊆ {a}×A⊥ .
Then every hyperplane
H
separating x from a separates two parallel
A, since A ⊆ B . It follows from Lemma 1.9
{a} × A⊥ , whence x ∈ {a} × A⊥ .
and thus separates two parallel copies of
that every hyperplane separating
a
from
x
crosses
F ⊆ X , x a base 0cube f ∈ F and for brevity, let F ⊥ =
⊥
⊥
{f } ×
⊆ X . Note that f ∈ F ⊥ , and so we let F ⊥ = {f } × {f } × F ⊥ , which again
⊥ ⊥
contains f , and so we can similarly dene
F⊥
etc. In many other contexts, the orthogonal
Given a convex subcomplex
F⊥
complementation operation is an involution, but this is not true in cube complexes. However:
Lemma 3.2.
Let F be a convex subcomplex of X . Then
F⊥
⊥ ⊥
= F ⊥.
Proof. If F is a convex subcomplex, there is a parallel copy of F ⊥ based at each 0cube of F ,
⊥ is a convex subcomplex of X . Thus F ,→
⊥ ⊥ , and by Lemma 3.1 we have
since F × F
k F
⊥ ⊥
F⊥ ⊇
F⊥
. To obtain the other inclusion, we show that every parallel copy of F is
⊥ ⊥ . This is clear since, letting A = F ⊥ , we have that A⊥ × A
contained in a parallel copy of F
⊥
⊥ ⊆ F ⊥ ⊥ × F ⊥ , both
is a convex subcomplex of X , but F ⊂ A by the above, and thus F × F
⊥ ⊥
⊥ ⊆
of which are convex subcomplexes of X . Hence F
F⊥
, completing the proof.
Lemma 3.3.
Let {Ai }i∈I be a set of convex subcomplexes of X with diam(Ai ) > 0 for all i and
⊥
S
S
choose ai ∈ Ai . Then for any a ∈ Hull i∈I Ai , we have that {a} × Hull i∈I Ai
is the
⊥
largest subcomplex parallel into {ai } × Ai for all i ∈ I .
HIERARCHICAL HYPERBOLICITY OF ALL CUBICAL GROUPS
Further, if a ∈
Proof.
T
i∈I
Ai 6= ∅, then {a} × Hull
V
(1)
(2)
i∈I
Ai
⊥
=
T
i∈I
{a} × Ai ⊥ .
To prove the rst assertion, we must show that the hyperplanes crossing
are precisely the hyperplanes crossing each
If
S
is a hyperplane which crosses
V
V
crosses
Ai
separates
Hull
⊥
A
Si
for all
i∈I Ai
i ∈ I.
V
, then
11
Hull
S
i∈I
Ai
⊥
has one of two types. Either:
i;
Aj for i, j ∈ I ;
for some
Ai
from
If H is a hyperplane which crosses all such V , then for all i and any ai ∈ Ai , H crosses
{ai } × Ai ⊥ , since it crosses every hyperplane which crosses each Ai . Now, suppose H crosses
every hyperplane crossing each Ai and let V be a hyperplane which separates Aj and Ak . Then
V must separate every hyperplane crossing Aj from every hyperplane crossing Ak . S
Thus, H
must cross V . Thus, the set of hyperplanes crossing every hyperplane crossing Hull
i∈I Ai
⊥
is exactly the intersection of the hyperplanes which cross each {ai } × Ai , completing the proof.
The second statement follows immediately from the above discussion and Denition 1.5. Alternative characterization of F.
For convex subcomplexes D, F of X , we write F =
F = {f } × D⊥ for some f ∈ F . Let F be the hyperclosure in X , and assume for
the rest of the section that X is uniformly locally nite, i.e. there exists B < ∞ so that each
0cube x of X lies in ≤ B combinatorial hyperplanes.
3.1.
D⊥
to mean
Lemma 3.4.
Let H be an innite set of hyperplanes in the uniformly locally nite CAT(0)
cube complex X . Suppose that, if H, H 0 ∈ H and V is some hyperplane separating H, H 0 , then
V ∈ H. Then there exists a sequence (Hn )n≥0 of hyperplanes in H such that Hn separates Hn−1
from Hn+1 for all n ≥ 1.
Proof. For each n ≥ 1, choose a nite subset Hn of H so that:
• H1 consists of a single hyperplane H1 ;
• Hn ( Hn+1 for each n ≥ 1;
• if H, H 0 ∈ Hn and V is some hyperplane separating H, H 0 , then V ∈ Hn .
0
(0) so that every
Choose a 0cube x0 ∈ N (H1 ). For each n ≥ 1, let Xn be the set of all xn ∈ X
0
hyperplane separating xn from x0 belongs to Hn . Since Hn is closed under separation, Xn 6= ∅
0
and moreover, for each H ∈ Hn , there exists y ∈ Xn so that H separates y from x0 . Let Xn
0
be the convex hull of Xn ∪ {x0 }. Observe that Xn ( Xn+1 for all n. Moreover, since there is
a uniform bound on the degree of each 0cube in X , and hence in each Xn , and the number
n
of hyperplanes crossing Xn tends to innity as n → ∞, we see that diam(Xn ) −→ ∞. König's
lemma provides an innite geodesic ray in X , all of whose hyperplanes belong to ∪n Hn ⊆ H.
The nite dimension of X and Ramsey's theorem allow one to choose the desired (Hn )n≥1 from
among the hyperplanes crossing this ray.
Theorem 3.5
F ⊂X
(1)
(2)
(3)
(Alternative characterization of F). Let X be uniformly locally nite and let
be a convex subcomplex. Then the following are equivalent:
there exists a convex subcomplex A so that F = A⊥ ;
there exists S
a compact convex subcomplex C so that F = C ⊥ ;
F ∈ {X } ∪ n Fn .
Example 3.6 (Degenerate cases of Theorem 3.5).
A = X , then F = A⊥ is a 0cube, which,
being an intersection of combinatorial hyperplanes, lies in ∪n Fn . But the 0cube F is also easily
seen to be the orthogonal complement of any maximal cube C containing it. On the other hand,
⊥
if A = C is a 0cube, then F = A = X .
If
Proof of Theorem 3.5. Clearly (2) implies (1). We now prove that (3) implies (2). Let F ∈ Fn
⊥ for some 1cube e of X .
for n ≥ 1. If n = 1 and F is a combinatorial hyperplane, F = e
HIERARCHICAL HYPERBOLICITY OF ALL CUBICAL GROUPS
12
F = gH (F 0 ) where F 0 ∈ Fn−1 and H is a combinatorial
0
0 ⊥
0
hyperplane. Induction on n gives F = (C ) for some compact convex subcomplex C .
0
Let e be a 1cube with orthogonal complement H ∈ F1 , chosen as close as possible to C , so
0
0
0
that d(e, C ) = d(H, C ). Let C be the convex hull of (the possibly disconnected set) e ∪ C . By
⊥ is the largest convex subcomplex of X parallel into
Lemma 3.3 if we x x = H ∩ e, {x} × C
H and F 0 containing x. However, that is exactly gH (F 0 ). Moreover, since only nitely many
hyperplanes V appear on the above list, C is compact.
⊥ consists of a single
We now prove (1) =⇒ (3). Let A ⊆ X be a convex subcomplex. If A
point then there is nothing to prove, since each 0cube is a nite intersection of combinatorial
⊥
hyperplanes and thus lies in ∪n Fn . If A = X , then we are done by hypothesis.
Hence suppose that A is not unique in its parallelism class and diam(A) > 0 (otherwise,
A⊥ = X ). Let P ∼
= A × A⊥ be the convex hull in X of the union of all parallel copies of A,
(0) . Let H , . . . , H be all of the hyperplanes intersecting A whose carriers contain
x x ∈ A
1
k
x, so that for each i, there is a combinatorial hyperplanes Hi+ associated to Hi with x ∈ Hi+ .
+
k
Let Y = ∩i=1 Hi , which is a convex subcomplex such that Y ∩ A = {x}. Let S be the set
0
of all combinatorial hyperplanes associated to hyperplanes crossing A. Let H ∈ S . Since
gY (H 0 ∩ A) ⊆ Y ∩ A = {x}, we see that x ∈ gY (H 0 ). We claim that
\
A⊥ =
gY (H 0 ),
Next, assume that
n≥2
and write
H 0 ∈S
A⊥
A at x.
y ∈ A⊥ . Then every hyperplane V separating y from x crosses
0
0
0
each of the hyperplanes H crossing A, and thus crosses Y , whence y ∈ gY (H ) for each H ∈ S .
T
T
0
⊥
0
Thus A
⊆ H 0 ∈S gY (H ). On the other hand, suppose that y ∈ H 0 ∈S gY (H ). Then every
0
⊥
hyperplane H separating x from y crosses every hyperplane crossing A, so y ∈ A .
Second, the preceding assertion shows that it is sucient to product a nite collection H of
0
hyperplanes H crossing A so that
\
\
gY (H 0 ) =
gY (H 0 ).
where
denotes the orthogonal complement of
To prove this, suppose that
H 0 ∈S
H 0 ∈H
Indeed, if there is such a collection, then we have shown
⊥
many elements of Fk , whence A
for any nite collection
H ⊂ S,
∈ Fk+|H| ,
A⊥
to be the intersection of nitely
as required. Hence suppose for a contradiction that
we have
\
H 0 ∈S
gY (H 0 ) (
\
gY (H 0 ).
H 0 ∈H
m, let Hm be the (nite) set of hyperplanes H 0 intersecting A ∩ Nm (x) (and hence
0
satisfying x ∈ gY (H )). Observe that Hm is closed under separation in the sense that if V is a
hyperplane separating two elements of Hm , then V ∈ Hm .
Consider the collection Bm of all hyperplanes W such that W crosses each element of Hm and
W crosses Y , but W fails to cross A⊥ . (This means that there exists j > m and some U ∈ Hj
so that W ∩ U = ∅.)
S
Note that either
m≥1 Bm is nite, or it satises the hypotheses of Lemma 3.4. In the
former case, there exists m so that we can take Hm to be our desired set H, and we are done.
Hence suppose we are in the latter case, so that Lemma 3.4 produces a sequence (Wn )n≥0 of
hyperplanes in ∪m Bm so that Wn separates Wn±1 for n ≥ 1.
S
For each n, there is a hyperplane Hn ∈
m≥1 Hm so that Wn separates Hn from W1 . Indeed,
choose Hn so that Wn fails to cross Hn . Since Hn crosses some ball about x in A, for each
suciently large k , we have that Wk crosses Hn . Now, if k > n, since Wn separates Wk , W1 ,
and Wn does not cross Hn , and Wk crosses Hn , we must have that Wn separates Hn , W1 . Note
For each
HIERARCHICAL HYPERBOLICITY OF ALL CUBICAL GROUPS
13
−→
−→
g (H ) ⊆ Wn ∩ Y , where Wn is the halfspace associated to Wn that contains Hn . Thus
TY n
T −→
x ∈ n gY (Hn ) ⊆ n Wn , which is an innite decreasing sequence of halfspaces, which is thus
empty. This is a contradiction.
that
Corollary 3.7.
Let A be a convex subcomplex of X . There exists a compact convex subcomplex
C ⊆ A so that C ⊥ = A⊥ .
Proof.
Given a convex subcomplex
A,
Theorem 3.5 guarantees that
A⊥ ∈ F,
and is, in fact, the
A onto a specied
A⊥ is precisely the orthogonal complement
of the convex hull of a nite set of edges, each of which can be chosen in A, dual to a hyperplane
in H. Let C be this convex hull. Then C is convex, compact, and C ⊆ A, as desired.
H
intersection of the projections of a nite collection
hyperplane
H , crossing A.
of hyperplanes crossing
However, this implies that
4. Proof of the main theorem
We are now ready to prove our main result. Actually, our proof shows that in any uniformly
locally nite CAT(0) cube complex, the hyperclosure does not contain an innite ascending
chain that does not stabilize.
Proof of Theorem B. Let X be a proper CAT(0) cube complex with a proper, cocompact action
by a group G. Let F be the hyperclosure; our goal is to prove that there exists N < ∞ so
that each 0cube of X is contained in at most N elements of F. Recall from Corollary 2.2 that
F = ∪n≥1 Fn , where the Fn are dened inductively as in Section 2.1.
Lemma 2.8 shows that either the multiplicity of F is nite, or there exists a 0cube x ∈ X
and a sequence (Fi )i≥1 in F so that x ∈ Fi ( Fi+1 for each i ≥ 1. For the sake of brevity,
⊥ denote the orthogonal complement of E based at x. Now,
given any subcomplex E 3 x, let E
Corollary 3.7 provides, for each i ≥ 1, a compact, convex subcomplex Ci , containing x and
⊥
⊥
contained in Fi , so that Ci = Fi .
0
0
0
0
Let C1 = C1 and, for each i ≥ 2, let Ci = Hull(Ci−1 ∪ Ci ). Then Ci is convex, by denition,
0
and compact, being the convex hull of the union of a pair of compact subspaces. Now, Ci ⊆ Fi
0
0
for each i. Indeed, C1 = C1 ⊆ F1 by construction. Now, by induction, Ci−1 ⊂ Fi−1 , so
0
0
Ci−1
⊂ Fi since Fi−1 ⊂ Fi . But Ci ⊂ Fi by construction, so Ci ∪ Ci−1
⊂ Fi , whence the hull of
0
0
that union, namely Ci , lies in Fi since Fi is convex. Hence Ci ⊆ Ci ⊆ Fi for i ≥ 1.
⊥
0 ⊥
⊥
⊥
0
0
By Lemma 3.1, we have for each i that Fi ⊆ (Ci ) ⊆ Ci = Fi since Ci ⊆ Ci . Hence (Ci )i≥1
⊥
0
⊥
is an ascending sequence of convex, compact subcomplexes, all containing x, with (Ci ) = Fi
0 ⊥
⊥
for all i. By Lemma 3.1, ((Ci ) )i≥1 = (Fi )i≥1 is a descending chain. Further, we must have
⊥
⊥
that Fi
6= Fi+1 for all i. To see this, recall from Theorem 3.5 that there exists a convex
⊥
⊥
⊥
subcomplex Ai , which can be chosen to contain x so that Ai = Fi . Thus, if Fi = Fi+1 , then
⊥ ⊥
Fi ⊥
= Fi+1 ⊥ .
Using Lemma 3.2 this means that
Let
T∞ =
T
Fi = Fi+1 .
0 ⊥
i (Ci ) . By Lemma 3.3,
there is a compact convex subcomplex
since
D∞
is compact,
0
D ∞ ( CR
Thus, the sequence
T∞ =
D∞
0
i≥1 Ci
⊥
(Fi ⊥ )i≥1
is
strictly
decreasing.
. It follows from Corollary 3.7 that
S
⊥ and D
0
T∞ = D∞
∞ ⊂
i≥1 Ci . However,
0 )⊥ ( D ⊥ = T , a
R ≥ 1. Hence T∞ ( (CR
∞
∞
so that
for some
contradiction, completing the proof.
S
5. Factor systems and the simplicial boundary
Corollary C follows from Theorem B, Proposition 5.1 and [Hag13, Lemma 3.32].
Proposi-
tion 5.1 is implicit in the proof of [DHS16, Theorem 10.1]; we give a streamlined proof here.
HIERARCHICAL HYPERBOLICITY OF ALL CUBICAL GROUPS
14
Proposition 5.1. Let X
be a CAT(0) cube complex with a factor system S. Then each simplex σ
of ∂4 X is visible, i.e. there exists a combinatorial geodesic ray α such that the set of hyperplanes
intersecting α is a boundary set representing the simplex σ .
Proof. Let σ be a simplex of ∂4 X . Let σ 0 be a maximal simplex containing σ , spanned by
v0 , . . . , vd . The existence of σ 0 follows from [Hag13, Theorem 3.14]. By Theorem 3.19 of [Hag13],
σ 0 is visible, i.e. there exists a combinatorial geodesic ray γ such that the set H(γ) of hyperplanes
0
crossing γ is a boundary set representing σ . We will prove that each 0simplex vi is visible. It
0
will then follow from [Hag13, Theorem 3.23] that any face of σ (hence σ ) is visible.
Let Y be the convex hull of γ . The set of hyperplanes crossing Y is exactly H(γ). Since Y is
convex in X , Lemma 8.4 of [BHS14] implies that Y contains a factor system.
Fd
By Theorem 3.10 of [Hag13], we can write H(γ) =
i=1 Vi , where each Vi is a minimal
boundary set representing the 0simplex vi . Moreover, up to reordering and discarding nitely
many hyperplanes (i.e. moving the basepoint of γ ) if necessary, whenever i < j , each hyperplane
H ∈ Vj crosses all but nitely many of the hyperplanes in Vi .
i
For each 1 ≤ i ≤ d, minimality of Vi provides a sequence of hyperplanes (Vn )n≥0 in Vi so
i
i
i
i
that Vn separates Vn±1 for n ≥ 1 and so that any other U ∈ Vi separates Vm , Vn for some
m, n (one may have to discard nitely many hyperplanes from Vi for this to hold, but this is
unproblematic: it simply replaces γ with a sub-ray and shrinks Y correspondingly).
We will show that, after discarding nitely many hyperplanes from H(γ) if necessary, every
element of Vi crosses every element of Vj , whenever i 6= j . Since every element of Vi either lies
i
in (Vn )n or separates two elements of that sequence, it follows that U and V cross whenever
T
U ∈ Vi , V ∈ Vj and i 6= j . Then, for any i, choose n ≥ 0 and let H = j6=i Vnj . Projecting γ to
H yields a geodesic ray in Y , all but nitely many of whose dual hyperplanes belong to Vi , as
j
i
required. Hence it suces to show that Vn and Vm cross for all m, n whenever i 6= j .
j
i
Fix j ≤ d and i < j . For each n ≥ 0, let m(n) ≥ 0 be minimal so that V
m(n) fails to cross Vn .
Vmj , then, since Vmj crosses all
i
i
but nitely many of the hyperplanes from Vi , it crosses Vk for k >> n. Since it also crosses Vn ,
i
i
it must also cross Vr for all n ≤ r ≤ k . By discarding Vk for k ≤ n we complete the proof. Now
j
i
suppose that m(n) is bounded as n → ∞. Then there exists N so that Vn , Vm cross whenever
m, n ≥ N , and we are done, as before.
Hence suppose that m(n) → ∞ as n → ∞. In other words, for all m ≥ 0, there exists
n ≥ 0 so that Vmj crosses Vki if and only if k ≥ n. Choose M 0 and choose n maximal with
j
i
i
i
m(n) < M . Then all of the hyperplanes Vm(k)
with k ≤ n cross Vk , . . . , Vn but do not cross Vt
Note that we may assume that this is dened: if
Vni
crosses all
j
t < k . Hence the subcomplexes gVni (Vki ), k ≤ n are all dierent: gVni (Vki ) intersects Vm(k)
but
i
i
i
i
gVni (Vk−1 ) does not. On the other hand, since Vk separates V` from Vn when ` < k < n, every
i
i
i
i
i
hyperplane crossing Vn and V` crosses Vk , so gV i (Vk ) ∩ gV i (V` ) 6= ∅. Thus the factor system on
n
n
Y has multiplicity at least n. But since m(n) → ∞, we could choose n arbitrarily large in the
preceding argument, violating the denition of a factor system.
for
Proof of Corollary D.
γ is a CAT(0) geodesic, then it can be approximated, up to Hausdor
dim X , by a combinatorial geodesic, so assume that γ is a combinatorial
geodesic ray. By Corollary C, the simplex of ∂4 X represented by γ is spanned by 0simplices
v0 , . . . , vd with each vi represented by a combinatorial geodesic ray γi . The proof
Q of [Hag13,
Theorem 3.23] shows that, up to truncating a nite segment from γ , we have Y ∼
= i Yi , where
Yi = Hull(γi ). The projection of γ to each γi is a geodesic representing vi . The product of these
geodesics is a combinatorially isometrically embedded (possibly non-convex) (d+1)-dimensional
orthant subcomplex of Y containing (the truncated) γ .
distance depending on
If
HIERARCHICAL HYPERBOLICITY OF ALL CUBICAL GROUPS
15
References
[BH99]
[BH16]
[BHS14]
[BHS15]
[BKS08]
[BW12]
[Che00]
[CS11]
[DHS16]
[Hag07]
[Hag13]
[Hag14]
[Hua14]
[HW15]
[MM99]
[MM00]
[NR97]
[NR98]
[OW11]
[Sag95]
[Sag14]
[SW05]
[Wis]
[Wis04]
[Xie05]
Martin R. Bridson and André Haeiger, Metric spaces of non-positive curvature, Grundlehren der
Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, SpringerVerlag, Berlin, 1999. MR 1744486 (2000k:53038)
Jason Behrstock and Mark F Hagen, Cubulated groups: thickness, relative hyperbolicity, and simplicial
boundaries, Groups Geom. Dyn. 10 (2016), no. 2, 649707. MR 3513112
Jason Behrstock, Mark F Hagen, and Alessandro Sisto, Hierarchically hyperbolic spaces I: curve complexes for cubical groups, arXiv:1412.2171 (2014).
, Hierarchically hyperbolic spaces II: combination theorems and the distance formula,
arXiv:1509.00632 (2015).
Mladen Bestvina, Bruce Kleiner, and Michah Sageev, Quasiats in CAT(0) complexes, arXiv preprint
arXiv:0804.2619 (2008).
Nicolas Bergeron and Daniel T. Wise, A boundary criterion for cubulation, Amer. J. Math. 134 (2012),
no. 3, 843859. MR 2931226
Victor Chepoi, Graphs of some CAT(0) complexes, Advances in Applied Mathematics 24 (2000), no. 2,
125179.
Pierre-Emmanuel Caprace and Michah Sageev, Rank rigidity for CAT(0) cube complexes, Geom. Funct.
Anal. 21 (2011), no. 4, 851891. MR 2827012
Matthew G Durham, Mark F Hagen, and Alessandro Sisto, Boundaries and automorphisms of hierarchically hyperbolic spaces, arXiv preprint arXiv:1604.01061 (2016).
Frédéric Haglund, Isometries of CAT(0) cube complexes are semi-simple, arXiv preprint
arXiv:0705.3386 (2007).
Mark F Hagen, The simplicial boundary of a CAT(0) cube complex, Algebraic & Geometric Topology
13 (2013), no. 3, 12991367.
, Weak hyperbolicity of cube complexes and quasi-arboreal groups, J. Topol. 7 (2014), no. 2,
385418. MR 3217625
Jingyin Huang, Top dimensional quasiats in CAT(0) cube complexes, arXiv:1410.8195 (2014).
Mark F. Hagen and Daniel T. Wise, Cubulating hyperbolic free-by-cyclic groups: the general case, Geom.
Funct. Anal. 25 (2015), no. 1, 134179. MR 3320891
Howard A. Masur and Yair N. Minsky, Geometry of the complex of curves. I. Hyperbolicity, Invent.
Math. 138 (1999), no. 1, 103149. MR 1714338 (2000i:57027)
H. A. Masur and Y. N. Minsky, Geometry of the complex of curves. II. Hierarchical structure, Geom.
Funct. Anal. 10 (2000), no. 4, 902974. MR 1791145
Graham Niblo and Lawrence Reeves, Groups acting on CAT(0) cube complexes, Geom. Topol. 1 (1997),
approx. 7 pp. (electronic). MR 1432323
G. A. Niblo and L. D. Reeves, The geometry of cube complexes and the complexity of their fundamental
groups, Topology 37 (1998), no. 3, 621633. MR 1604899
Yann Ollivier and Daniel T. Wise, Cubulating random groups at density less than 1/6, Trans. Amer.
Math. Soc. 363 (2011), no. 9, 47014733. MR 2806688
Michah Sageev, Ends of group pairs and non-positively curved cube complexes, Proc. London Math.
Soc. (3) 71 (1995), no. 3, 585617. MR 1347406
, CAT(0) cube complexes and groups, Geometric group theory, IAS/Park City Math. Ser., vol. 21,
Amer. Math. Soc., Providence, RI, 2014, pp. 754. MR 3329724
Michah Sageev and Daniel T. Wise, The Tits alternative for CAT(0) cubical complexes, Bull. London
Math. Soc. 37 (2005), no. 5, 706710. MR 2164832
Daniel T Wise, The structure of groups with a quasiconvex hierarchy. Preprint (2011).
, Cubulating small cancellation groups, Geometric & Functional Analysis GAFA 14 (2004), no. 1,
150214.
Xiangdong Xie, The Tits boundary of a CAT(0) 2-complex, Transactions of the American Mathematical
Society 357 (2005), no. 4, 16271661.
DPMMS, University of Cambridge, Cambridge, UK
E-mail address : [email protected]
Mathematics Department, University of Nebraska-Lincoln, Lincoln, Nebraska, USA
E-mail address : [email protected]