SIMPLICES AT INFINITY IN CAT(0) CUBE COMPLEXES
MARK F. HAGEN
These are notes for lectures given in Nice in March 2016, on simplicial boundaries
of CAT(0) cube complexes. They cover the denition, basic properties, some applications to
cubulated groups, and some open problems.
Abstract.
Contents
Motivation and outline
1
A
1
Question
Outline
2
Notes on exercises
3
Acknowledgments
3
1.
3
Part I: the simplicial boundary
1.1.
The denition
4
1.2.
Visibility
7
1.3.
Comparing the boundaries
1.4.
More exercises and problems from Part I
2.
8
10
Part II: Quasi-trees, rank-one stu, and hierarchies
11
2.1.
The contact graph
11
2.2.
Rank-one things, the contact graph, and the boundary
11
2.3.
Factor-systems and the factored contact graph
13
2.4.
Problems on Part 2
15
3.
Part III: The compact simplicial boundary, rank-rigidity etc., and Question
∂4 X
A
15
3.1.
Redening
3.2.
Stationary measures on the boundary, rank-rigidity, and the like
using factor systems
15
20
3.3.
Final problems
22
References
25
Motivation and outline
Question
A .
Here is an arcane-looking question that I hope to show is interesting:
A0 ). Suppose that the proper CAT(0) cube complex X admits a geometric
action by some group G. Consider the smallest collection F of convex subcomplexes so that:
(1) N (H) ∈ F for each hyperplane H , where N (H) denotes the carrier;
0
0
0
(2) if F, F ∈ F, then gF (F ) ∈ F, where gF (F ) is the convex subcomplex of F spanned by
0
the vertices of F which are closest-point projections on F of vertices of F .
Does there exist N ∈ N so that each x ∈ X lies in at most N elements of F?
Question (Question
These notes are a circuitous route to this question, via the
simplicial boundary ∂4 X
of
X,
which is a combinatorial analogue of the Tits boundary, introduced for quite dierent reasons.
Date :
September 1, 2016.
1
SIMPLICES AT INFINITY IN CAT(0) CUBE COMPLEXES
A positive answer to Question
A0
2
would have implications for the structure of
admits a proper, cocompact group action. Specically, we would be able to use
∂4 X when X
∂4 X to study
quasi-isometry invariants like thickness and divergence without introducing extra hypotheses.
This application of Question
A0
and this viewpoint on
∂4 X
are not the main focus of these
notes (but see the problems at the end).
A0
Instead, it turns out that when Question
tive description of
F.
∂4 X
has a positive answer, there is an alterna-
in terms of a collection of quasi-trees associated to the elements of
This is intimately related to the hierarchical structure of such cube complexes discussed
in [BHS14, BHS15], and this new way of looking at
∂4 X
is a special case of what can be done for
general hierarchically hyperbolic spaces [DHS15]. One consequence of this is that the simplicial
boundary can be retopologized in a fairly gentle way (i.e. the identity from
∂4 X to the newlyX that encodes
topologized thing is an embedding on simplices), yielding a compactication of
more data than the visual boundary or the Roller compactication. One can then do various
cool things. As an example, we'll consider
Gstationary
measures on this compactication to
give a new proof of the Caprace-Sageev rank-rigidity theorem [CS11] (in many cases).
A0 is reasonable because it has a positive answer for every known example. For
instance, if X is a compact nonpositively-curved cube complex admitting a local isometry to
0
the Salvetti complex SΓ of the right angled Artin group AΓ , for some graph Γ, then Question A
has a positive answer for the universal cover of X . Hence, here's a punchier question:
Question
Question (Question
A).
Let
compact action by some group
X → SeΓ
X be a proper CAT(0) cube complex admitting a proper, coG. Does there exist a nite graph Γ and a cubical embedding
whose image is a convex subcomplex?
(It's called Question
of the Salvetti complex
A
SΓ
A in [BHS15].) Here, SeΓ is the universal cover
Artin group AΓ . Observe that we are not asking
because it is Question
of the right-angled
for any kind of relationship between
G
and
AΓ ,
or any kind of equivariance of the map. This
question appears to be less nutty than it perhaps looks.
Outline. In Part 1 denes the simplicial boundary
∂4 X
of
X,
properties, and discusses how it compares to other boundaries.
covers
visibility,
which is about associating simplices in
∂4 X
describes some of its basic
This part of the notes also
to geodesic rays in
X.
Whether
visibility is guaranteed by geometric group actions is unknown and related to Question
A.
Part 2 is about rank-one phenomena from the boundary viewpoint. This is closely related
to another object associated to
X,
the
contact graph CX ,
which is the intersection graph of the
hyperplane-carriers. This graph is always a quasi-tree, on which
Aut(X )
acts [Hag14b]. The
simplicial boundary was actually introduced as a bookkeeping device in an attempt to answer
CX is bounded (as it is when, for instance, X splits as a nontrivial product).
Geodesic rays in X project to subgraphs in CX in a natural way, so one way to answer the
question of boundedness of CX is to look for a ray γ in X whose shadow in CX is unbounded.
This turns out to hold for any γ for which the obvious obstructions are absent: γ must be rank-
the question of when
one in the appropriate sense, and it must not lie uniformly close to any hyperplane. Rank-one
turns out to mean the same thing as γ represents an isolated point in
∂4 X . We'll state some
∂4 X , isometries of X ,
results, which make use of lemmas of Caprace-Sageev [CS11], that relate
X.
CX from a more sophisticated point of view.
Roughly, if γ fails to project quasi-geodesically to CX , then a more rened version of the above
analysis provides a collection of hyperplanes on which γ has large projection, and we can consider
(roughly) the projection of γ to the contact graphs of these hyperplanes, etc. To make all of this
work out technically, we need to impose conditions on X , which would be guaranteed to hold if
the answer to Question A were yes. More precisely, we can often associate to X a collection
their actions on
CX ,
and product decompositions of
We then re-examine the projection of
γ
to
SIMPLICES AT INFINITY IN CAT(0) CUBE COMPLEXES
of quasi-trees, the
factored contact graphs,
which completely encode the geometry of
3
X
(up
to quasi-isometry). Part 3 is about an alternative denition of the boundary in terms of the
boundaries of the factored contact graphs. This yields the compact version of the simplicial
boundary, and we close by discussing some uses of this object.
Notes on exercises. This text contains various exercises and problems designed to stimulate
discussion and/or build familiarity with the notions being used, as well as a few open problems.
The non-open problems are either very straightforward, or solutions to them are explained
1
in [Hag13, Hag14b] or the forthcoming paper [DHS15]. The open or tricky problems are starred.
Acknowledgments. I am very grateful to Indira Chatterji for organizing the meeting which
provided the opportunity to write these notes, and for her interest in the subject. Also, many
thanks to Mike Carr, Ruth Charney, Dave Futer, Dan Guralnik, Jingyin Huang, Alessandra
Iozzi, Brian Rushton, Michah Sageev, Yulan Qing, and Dani Wise for interesting discussions,
observations, and questions about the objects studied here. Part of the notes are based on joint
work with subsets of
{Jason Behrstock, Matt Durham, Alessandro Sisto}, with whom it's been
extremely eye-opening and enjoyable to collaborate.
1. Part I: the simplicial boundary
We start with the denition and fundamental properties of the simplicial boundary of a
CAT(0) cube complex, along with examples and pictures one should have in mind.
X is a CAT(0) cube complex, H is the set of hyperplanes in X . The crossing
H is: H, H 0 ∈ H cross, written H⊥ H 0 , if H ∩H 0 6= ∅. If H, H 0 ∈ H are not separated
by some third hyperplane, they contact, written H ^H
⊥ 0 . Note that crossing hyperplanes contact.
Throughout,
relation on
Figure 1. Some crossing and osculating hyperplanes. Hyperplanes contact if
they either cross or osculate.
Convention 1.1. Throughout these lectures, we make only one standing niteness assumption
about X : any set of pairwise-crossing hyperplanes is nite. This is weaker than requiring
X
to be nite-dimensional or locally nite. (In applications, we often impose one of the latter two
conditions, or the even stronger condition that
0cubes
of
X
have uniformly bounded degree.)
Remark 1.2 (Convexity). What really makes CAT(0) cube complexes so organized and so
nice as a setting for doing geometry is their hyperplanes, and the combinatorics of how the
hyperplanes interact.
One of the main ways that this niceness manifests is in the very rich
notion of convexity that the hyperplane structure provides.
1
Very soon, as of September 1, 2016
Whether one adopts the median
SIMPLICES AT INFINITY IN CAT(0) CUBE COMPLEXES
4
graph viewpoint, or uses disk diagrams over cube complexes, convexity usually lurks near the
crux of whatever one is doing.
X be a CAT(0) cube complex.
0skeleton appears in Y . A full
Let
whose
X (1) .
A subcomplex
Y
Y
is
subcomplex
full if Y contains every cube of X
convex if Y (1) is metrically convex in
is
There are numerous equivalent formulations.
Y ⊆ X is convex, then Y inherits a CAT(0) cubical structure from X , and the
hyperplanes of Y have the form H ∩ Y , where H is a hyperplane of X . Convexity of Y means
that any two hyperplanes of X that intersect Y intersect one another in X if and only if their
intersections with Y intersect one another.
Second, any subspace Y of X has a convex hull, which is just the intersection of all convex
subcomplexes containing Y . This has an attractive alternative characterization, in terms of
halfspaces. The archetypal convex subcomplex is the combinatorial hyperplane. A combinatorial
1
1 1
hyperplane is the image of H × {± } under the map H × [− , ] ∼
2
2 2 = N (H) ,→ X , where N (H)
denotes the carrier of H (i.e. the union of all closed cubes intersecting H ). A combinatorial
→
−
halfspace is dened as follows: let H be a halfspace associated to H (i.e. a component of
→
−
X − H ). There are two associated combinatorial halfspaces, namely H ∪ N (H) and the closure
First, if
of its complement; these are respectively bounded by the two dierent combinatorial hyperplanes
associated to
H.
A combinatorial halfspace is convex, and the convex hull of
of all combinatorial halfspaces containing
1.1.
Y
is the intersection
Y.
The denition. The simplicial boundary of
X
is, like the Tits boundary and the visual
boundary, designed to encode the dierent directions in which one can move o to innity
inside
X.
Instead of thinking about a space of geodesic rays, or innite sequences, we'll use
the hyperplanes to dene directions. First, consider a (graph-metric) geodesic ray
which is characterized by the property that each hyperplane intersects
H(γ)
The set
(1)
(2)
of hyperplanes
H
with
H ∩ γ 6= ∅
|H(γ)| = ∞;
H(γ) contains no facing triple :
if
γ
γ ⊂ X (1) ,
in at most one point.
has a few interesting properties:
H, H 0 , H 00 ∈ H(γ) are pairwise-disjoint, then one must
separate the other two;
(3)
(4)
closed under separation : if H, H 0 ∈ H(γ) are separated by some hyperplane V ,
then V ∈ H(γ);
H(γ) is unidirectional : if H ∈ H(γ), then at most one of the halfspaces associated to H
contains innitely many elements of H(γ).
H(γ)
is
Rather than dene a boundary in terms of geodesic rays, we work with sets of hyperplanes
modelled on
H(γ):
Denition 1.3 (Boundary set, almost-containment, equivalence, minimality). A
boundary set
is an innite set of hyperplanes that is unidirectional, closed under separation, and contains no
H, H0 , we say H0 almost contains H, written H H0 ,
|H −
∩ H| < ∞. If H H0 H, then H, H0 are equivalent. The boundary set H
minimal if H0 and H are equivalent whenever H0 H.
facing triple.
H0
Given boundary sets
H0 and
if
is
Example 1.4. Here are some examples of boundary sets:
γ is a geodesic ray in X , then H(γ) is a boundary set.
If X is a tree, then there is a bijection between the geodesic rays in X and the boundary
sets, given by γ → H(γ), which descends to a bijection between ∂X (Gromov boundary)
(1) If
(2)
and the set of equivalence classes of boundary sets.
X = α × β , where α, β are combinatorial
to H(α), H(β), or H(α) ∪ H(β).
(3) If
rays, then every boundary set is equivalent
SIMPLICES AT INFINITY IN CAT(0) CUBE COMPLEXES
X = A × B,
(4) More generally, if
X
H ,V
where
A, B
are CAT(0) cube complexes, then every
boundary set in
is equivalent to a boundary set in
H ∪ V,
are boundary sets of
where
A, B
5
A,
a boundary set in
B,
or a union
respectively.
[0, ∞)2 by 2cubes, let
f : [0, ∞) → [0, ∞) be some unbounded nondecreasing function with f (0) = 0, and let X
2
be the union of all closed 2cubes intersecting the set {(x, y) ∈ [0, ∞) | y ≤ f (x)}. This
is a staircase and is a major source of CAT(0) cube complex pathology. See Figure 2.
(5) This is a very important example: start with the obvious tiling of
Figure 2. A staircase. The set of all hyperplanes is a boundary set, in fact
equal to
H(α).
The set of vertical hyperplanes is a boundary set,
H(β). The
H(γ) for
set of horizontal hyperplanes is also a boundary set, but is not equal to
any geodesic ray
γ
(6) Next, we have the
where would such a ray start?
ziggurat.
Two types of cross-section hyperplane! are coarse rays
that look like staircases with plateaux, while the other type is a Euclidean quadrant.
In Figure 3, you can see a red boundary set, a blue boundary set, a pink boundary set.
Any union of two or more of these boundary sets is again a boundary set.
Figure 3. A ziggurat and its hyperplanes.
(7)
Clis.
The clis are to the ziggurat as the staircase is to the quadrant, in a way. See
Figure 4.
There is a red boundary set, whose hyperplanes are compact, and a blue
boundary set whose hyperplanes are staircases.
There is also a pink boundary set,
whose hyperplanes are products of rays with increasingly long intervals. The union of
all three sets is a boundary set, and the same is true for the union of blue and pink and
the union of the red and pink sets. However, the union of the red and blue sets is not
closed under separation! We will return to this example (which was pointed out by Dan
Guralnik and Alessandra Iozzi) later in the lecture.
Two facts about the structure of boundary sets are needed to dene the simplicial boundary:
Every boundary set contains a minimal
boundary set. If X is locally nite, then every innite subset of H contains a boundary set.
Proposition 1.5 (Finding minimal boundary sets).
SIMPLICES AT INFINITY IN CAT(0) CUBE COMPLEXES
6
Figure 4. Clis.
Proof.
See exercises.
Let H be a boundary set. Then
there exist pairwise-disjoint minimal boundary sets H0 , . . . , Hk so that:
(1) H is equivalent to tki=0 Hi ;
(2) for 0 ≤ i < j , the set Hj dominates Hi , i.e. for all H ∈ Hj , H crosses all but nitely
many elements of Hi .
0
Moreover, this decomposition is essentially unique: if H is equivalent to some other union tki=0 Hi0
of minimal boundary sets, then k = k 0 and, up to relabeling, Hi and Hi0 are equivalent.
Proposition 1.6 (Canonical decomposition of boundary sets).
Sketch.
H is minimal, we're done. Otherwise, nd a minimal subset H0 using Proposition 1.5.
H − H0 is nite, or contains an innite subset closed under separation; then
Proposition 1.5 to nd a minimal boundary set H1 ⊆ H − H0 . Figure 5 oers the hint on
If
Check that either
apply
how to do this.
Figure 5. The inductive step in the proof of Proposition 1.6. The red minimal
boundary set is given; any further innite subset of
H
behaves like one of the
blue ones.
Continue in this way.
To prove uniqueness, use the following fact: if
M, M0
boundary sets, then either they are equivalent or their intersection is nite...
The
k
dimension
of a boundary set (equivalence class of boundary sets)
in the decomposition of
H
H ([H])
are minimal
is the number
from Proposition 1.6.
simplicial boundary ∂4 X of X
k simplex for each k dimensional equivalence class of boundary
sets. Such a boundary simplex, a simplex corresponding to a boundary set H, is a face of the
0
0
simplex corresponding to H if H H .
Denition 1.7 (Boundary simplex, simplicial boundary). The
is the simplicial complex with a
SIMPLICES AT INFINITY IN CAT(0) CUBE COMPLEXES
7
Remark 1.8 (Warning). The example of clis shows that, while every maximal simplex of
∂4 X
is actually a boundary simplex corresponding to some equivalence class of boundary sets,
a simplex of
∂4 X
can have a proper face which does not correspond to an equivalence class of
boundary sets. See Exercise (2). However,
∂4 X
is the union of genuine boundary simplices,
and in practice one gets away with just thinking about these simplices. Under the assumption
of
full visibility
discussed below, this weirdness disappears.
Example 1.9 (First examples). Here are some examples to keep in mind.
(1)
∂4 X
X , when X is a tree.
∂4 X is a discrete set; see Exercise (5).
If X = A×B , then ∂4 X = ∂4 A?∂4 B , where ? denotes the simplicial join. This uses the
fact that product decompositions of X correspond to partitions of the set of hyperplanes
of X into disjoint sets V, H so that each element of V crosses each element of H.
The simplicial boundary of a staircase is a 1simplex; this is also the simplicial boundary
is a discrete set, equal to the set of ends of
(2) Suppose that
(3)
(4)
X
is hyperbolic. Then
we'd get if we lled in the rest of the quadrant.
X be the universal cover of the Salvetti complex of the Croke-Kleiner RAAG:
ha, b, c, d | [a, b], [b, c], [c, d]i. Then ∂4 X has many isolated 0simplices coming from
(5) Let
rank-one elements (more on this later), but the interactions between the standard ats
give rise to the subcomplex in Figure 6:
Figure 6. Eye of Sauron.
Theorem 1.10 (Basic properties).
∂4 X has the following properties:
(1) It is a ag complex.
(2) Every simplex of ∂4 X is contained in a nite-dimensional maximal simplex.
(3) Let Y ⊂ X be a convex subcomplex. Then there is a natural simplicial embedding ∂4 Y →
∂4 X whose image is a full subcomplex. This holds in particular when Y is the carrier
of a hyperplane.
Proof.
Assertion (1) follows from the denition. Assertion (2) follows from Proposition 1.6 and
the fact that there is no innite family of pairwise-crossing hyperplanes.
Y have the form H ∩ Y ,
H ∩H 0 ∩Y =
6 ∅ if and only if H ∩H 0 6= ∅,
It remains to prove assertion (3). By convexity, the hyperplanes of
where
H
is a hyperplane of
whenever
H, H 0
k dimensional
intersecting
Y.
Moreover,
Y (again by convexity). It follows that each
{H ∩ Y | H ∈ H}, where H is a k dimensional
are hyperplanes both intersecting
boundary set of
boundary set in
1.2.
X
X,
Y
has the form
and the claim follows.
Visibility. The denition of a boundary set was motivating by looking at the set of hy-
perplanes crossing a combinatorial geodesic ray. How far away can the denition wander from
this example? To what extent is
∂4 X
really a space of directions in
X?
SIMPLICES AT INFINITY IN CAT(0) CUBE COMPLEXES
8
h ⊆ ∂4 X is visible if
H(γ) represents h (i.e. h is the boundary simplex corresponding
0
to the equivalence class of H(γ)). In this case, we say that γ represents h. The pair h, h is visible
if there is a bi-innite geodesic γ in X which is the union of two rays with bounded intersection
0
that respectively represent h, h . The cube complex X is fully visible if every simplex is visible.
Denition 1.11 (Visible simplex, visible pair, full visibility). The simplex
there is a geodesic ray
γ
so that
Remark 1.12. Given a subcomplex
Y⊆X
A ⊆ ∂4 X , one could say A is visible if there is a subspace
A. I haven't thought about this general notion.
whose convex hull has boundary
Full visibility, and the failure thereof, is not fully understood; see Problem 1.3.1.(5).
Proposition 1.13.
Sketch.
Every maximal simplex of ∂4 X is visible.
∂4 X , so that m is represented by a boundary set M.
H(γ) = M; if we fail, it's going to reect that m lies
in some larger simplex. First, let M = {Hi }i≥0 be numbered so that i < j < k whenever Hj
separates Hi from Hk . For each i, nd a geodesic segment γi so that the set of hyperplanes
crossing γi is exactly {H0 , . . . , Hi }. This is possible because M is closed under separation and
Let
m
be a maximal simplex of
We will attempt to build a ray
γ
with
because of our numbering scheme.
Yi be the convex
γ0 ⊂ γ1 ⊂ . . ., and
Now let
can nd
large
i, j ,
Hi
Proposition 1.14.
Sketch.
H separating Yi , Yj . In this way, we build a sequence of
M ∈ M crosses all but nitely many Hi . Result: M is contained
large boundary set, contradicting maximality of m.
= ∅.
Represent
Let h, h0 be visible simplices of ∂4 X . Then h, h0 is a visible pair if and only
h, h0
γ, γ 0 with a common initial point. If γ, γ 0 only have nitely
0
(i.e. if h ∩ h = ∅), then there is some folding and truncation one
0
geodesic. Otherwise, h ∩ h 6= ∅.
by rays
many common hyperplanes
can do to build a bi-innite
1.3.
Check that either we
so that each
in, but not equivalent to, a
if h ∩
γi .
we can nd a hyperplane
hyperplanes
h0
hull of the union of all possible choices of
thus build a limiting ray, or the following happens: for arbitrarily
Comparing the boundaries. The denition of
Roller boundary, but in practice,
∂4 X
∂4 X
looks more like the denition of the
looks more like the Tits boundary. The following exercises
discuss these comparisons. (This section is a series of exercises rather than real exposition since
it's interesting but not relevant to my propaganda mission about Question
A.)
The simplicial boundary and the Tits boundary. Assume that X is fully visible.
(1) Let v be a simplex of ∂4 X , spanned by 0simplices v0 , . . . , vk . Show that for each i,
we can choose a combinatorial geodesic ray γi representing vi so that there is a cubical
Q
isometric embedding
i Ci → X , where Ci denotes the convex hull of γi .
(2) Prove that for each i, there is a CAT(0) geodesic ray αi , with the same initial point as
γ
inside
Qi , that crosses exactly those hyperplanes crossed by γi . Deduce that, somewhere Q
C
,
there
is
a
isometrically
embedded
(in
the
CAT(0)
sense)
Euclidean
orthant
i
i
i αi .
(3) Produce an embedding ∂4 X → ∂T X (the Tits boundary for the usual CAT(0) metric)
1.3.1.
sending each simplex to a right-angled spherical simplex.
∗
(4) What is the failure of your embedding to be surjective?
parameterize the embeddings
∗
∂4 X → ∂T X
Explain how the various
(5) Later, we'll see a condition that guarantees full visibility. For the moment: let
be the set of hyperplanes of
so that, if
m < n,
then
Hn
that the following holds:
image of
N (Hm )
Ci
(so, intersections of hyperplanes of
does not separate
Ci
of the type you just constructed.
Hm
X
with
{Hn }n≥1
Ci ), numbered
γi . Suppose
from the initial point of
consider all subcomplexes you can build by examining the
under combinatorial closest-point projection to
N (Hn ),
as
m, n
vary.
Suppose that the resulting collection of subcomplexes has bounded multiplicity. Then,
SIMPLICES AT INFINITY IN CAT(0) CUBE COMPLEXES
if this holds for each
0simplex of ∂4 X , X is fully visible. Also, by considering Figure 7,
∂4 X → ∂T X you constructed is surjective?
can you characterise when the map
Figure 7.
9
∂4 X → ∂T X
can fail to be surjective (left picture). The bottlenecks
in the right picture disallow that sort of behaviour.
SIMPLICES AT INFINITY IN CAT(0) CUBE COMPLEXES
1.4.
10
More exercises and problems from Part I.
(1) Prove Proposition 1.5. (Idea: given a boundary set
U , use the fact that sets of pairwiseH0 , H1 , . . . ∈ U with Hi separating
crossing hyperplanes are nite to produce a sequence
(2)
∗
Hi±1 for i ≥ 1. Now close this set under separation...)
Let γ be a combinatorial geodesic ray with convex hull C .
simplex, represented by γ .
∂4 C ⊆ ∂4 X
Then
(3) Why doesn't the bizarre phenomenon exhibited by clis happen when
X
is a single
is fully visible?
What can be said about the structure of the convex hull of a geodesic ray when
X
is
fully visible?
∗
(4) Just like the Tits boundary of a CAT(0) space can detect splittings of the CAT(0) space
as a metric product, the simplicial boundary can detect splittings as a cubical product.
One direction was seen in Example 1.9.
cube complex which is
essential
Conversely, let
X
be a fully visible CAT(0)
(i.e. every halfspace contains a hyperplane). Suppose
∂4 X ∼
= A ? B for nonempty subcomplexes A, B . Prove that X ∼
= XA × XB , where
XA and XB are unbounded convex subcomplexes. (Hint: x a base vertex x ∈ X and,
for each 0simplex v of ∂4 X lying in A, use full visibility to represent v by a ray γv
emanating from x. Let XA be the convex hull of the union of all of these rays, and
dene XB analogously...). Also: exhibit examples showing that the essentiality and full
that
visibility hypotheses are necessary.
(5) It is known [Hag14b] that, if
X
is uniformly locally nite, then
when the following holds for some integer
element of
that
∗
∂4 X
V
crossing every element of
is a discrete set when
(6) Give nice conditions on
X
X
H,
is hyperbolic exactly
V, H of hyperplanes with every
min{|V|, |H|} ≤ q . Deduce from this
for any sets
we have
is hyperbolic.
ensuring that it is fully visible.
a CAT(0) cube complex with a group
visible?
q:
X (1)
G
In particular:
given
acting properly and cocompactly, is
(Behrstock and I guess yes, in [BH, Conjecture 2.8].
X
X
fully
A positive answer to
this question would follow from a positive answer to a further-reaching question about
cubical groups that we'll discuss later.)
(7) Do the folding and truncation from Proposition 1.14, as suggested by Figure 8.
Figure 8. Folding and truncating to make a bi-innite geodesic: go beyond
the last common hyperplane.
(8) If
γ
γ0
is a geodesic
γ fellow-travel.
Gessential core.
Prove that
is a combinatorial geodesic ray whose convex hull is hyperbolic, and
0
ray representing the same boundary simplex, then γ and
G act geometrically
∂4 X = ∂4 Y .
(9) Let
on
X
and let
Y ⊆ X
be the
SIMPLICES AT INFINITY IN CAT(0) CUBE COMPLEXES
11
2. Part II: Quasi-trees, rank-one stuff, and hierarchies
2.1.
The contact graph. Given a CAT(0) cube complex
X , the contact graph CX
is the graph
with a vertex for each hyperplane and an edge joining two vertices if and only if the corresponding
hyperplanes contact. In other words,
CX
is the intersection graph of the hyperplane-carriers.
The contact graph is quasi-isometric to the space obtained from
X (1) by coning o the 1skeleton
of each hyperplane carrier.
Remark 2.1 (Mapping class group digression). In many ways, the contact graph is to the cube
complex as the curve graph of a surface is to the marking complex/mapping class group, so, if
you're familiar with the Masur-Minsky approach to the mapping class group [MM99, MM00], it
might help to have the curve graph in the back of your mind when thinking about the contact
graph. In fact, the relationship between the contact graph and the simplicial boundary allows
one to build a boundary for the mapping class group [BHS14, BHS15, DHS15].
Remark 2.2 (Contact graph facts). Some basic facts about the contact graph:
(1) If
Y⊆X
way:
is a convex subcomplex, then
each hyperplane
hyperplane
(2) If
A, B
join.
H
of
X
H∩Y
of
Y
CY
is an induced subgraph of
CX
in the obvious
CY ) corresponds to
CX ).
C(A × B) = CA ? CB , where ? denotes
(representing a vertex of
the
(and thus to the corresponding vertex of
are CAT(0) cube complexes, then
the
Indeed, every hyperplane crossing one factor crosses every hyperplanes crossing
the other factor. The CAT(0) cube complexes we have in mind have horrically locally
innite contact graphs, in general, because they can have unbounded hyperplanes, but
this example shows that the contact graph can nonetheless be bounded.
(3) Each
0cube x ∈ X
corresponds to a clique in
hyperplanes whose carriers contain
maximum degree of a vertex in
Theorem 2.3 ([Hag14b]).
x.
CX
whose vertices correspond to the
Hence the clique number of
CX
is equal to the
X.
CX is quasi-isometric to a tree.
In view of Theorem 2.3 and Remark 2.2.(2), it's natural to ask when
CX
is a quasi-point, i.e.
a quasi-tree by virtue of being bounded. In fact, this question motivated the denition of the
simplicial boundary, and it's closely related to rank-rigidity.
2.2.
Rank-one things, the contact graph, and the boundary. Let
geodesic ray in
X.
What does
like (intrinsically in
X
γ
γ
be a combinatorial
look like? More precisely, what does the convex hull of
γ
look
and from the point of view of the simplicial boundary)?
ith 1cube of γ . The sequence H1 , H2 , H3 , . . .
denes an edge-path in CX , since Hi ^H
⊥ i+1 for each i. Here is the main question:
• When is {Hi } unbounded? In other words, when does projecting γ to CX prove that
CX is an unbounded quasi-tree? Even better, when is γ(i) 7→ Hi a quasi-isometric
For
i ≥ 1,
let
Hi
be the hyperplane dual to the
embedding?
The following is proved in Section 2 of [Hag13]. We'll skip the proof here, and instead focus
on a more quantitative version of the same statement, which is more obviously related to what
we'll do in the next section.
Let γ be a combinatorial geodesic ray in X . Then
one of the following holds:
(1) γ lies in a uniform neighborhood of some hyperplane of X ;
(2) γ lies in an isometrically embedded staircase in X ;
(3) the set of hyperplanes intersecting γ span an unbounded subset of CX .
(The rst two conclusions can both hold.) In particular, γ projects to an unbounded subset in
CX only if it represents an isolated 0simplex in ∂4 X .
Theorem 2.4 (The projection trichotomy).
SIMPLICES AT INFINITY IN CAT(0) CUBE COMPLEXES
12
In fact, Theorem 2.2 of [Hag13] is stronger; paraphrased, it says:
Let γ be a combinatorial
geodesic ray in X . Suppose that there exists M so that for all hyperplanes H of X , there are at
most M hyperplanes U such that U ∩ H 6= ∅ and U ∩ γ 6= ∅. Then the subgraph of CX spanned
by the vertices corresponding to the set of hyperplanes crossing γ is quasi-isometric to γ .
Theorem 2.5 (The quantitative projection trichotomy [Hag13]).
Proof. Here's slicker version of the proof than you'll nd in [Hag13]. Let C be the convex hull of
γ in X , and let Λ be the subgraph of CX spanned by the set of hyperplanes crossing C (which is
0
0
the same as the set of hyperplanes crossing γ ). Let H, H cross C , and let H = H0 , . . . , Hn = H
be a geodesic sequence in CX joining them. Choose x ∈ N (H0 )∩C, y ∈ N (Hn )∩C to be 0cubes,
and let α0 α1 · · · αn−1 αn be a combinatorial path from x to y , where each αi is a combinatorial
geodesic in N (Hi ). Let β be a combinatorial geodesic from x to y (so β lies in C for example,
we could take x, y ∈ γ and have β be a subpath of γ , but it doesn't matter). See Figure 9.
Figure 9. The proof of Theorem 2.5.
Since
β −1 α0 · · · αn
is a closed path, it bounds a disc diagram
above be made so that, when we pull
D
D → X.
Let all the choices
tight, its area is as small as possible (so, as few squares
K be a dual curve in D
αi . If K ends on αj with |i−j| > 2, then we contradicted that Hi , Hi+1 , . . . , Hj
(or whatever) is a CX geodesic. If i = j , we contradict that αi is a geodesic. If |i − j| = 1,
then we can do some folding and remove a backtrack. If |i − j| = 2, then we can replace Hi+1
(say) by the hyperplane to which K maps and get a lower-area D . Hence all dual curves travel
between α0 · · · αn and β , i.e. α0 · · · αn is a geodesic starting and ending on C and thus lying
in C . In particular, each hyperplane crossing each αi crosses γ , so |αi | ≤ M for all i. Hence n
grows linearly in |β|, and we're done.
as possible and as few backtracks in the boundary path as possible). Let
starting on some
Remark 2.6. A similar argument yields the notion of
geodesics of
X
that track geodesics in
CX .
hierarchy paths in cube complexes
i.e.
It's shown in [BHS14, Section 3] that any two vertices
are joined by such a geodesic, using an argument basically identical to the proof of Theorem 2.5.
In the next section, we'll see that Theorem 2.5 is the tip of the iceberg in terms of understanding points in
∂4 X
in terms of projections to quasi-trees. First, we can state some related results
about isometries (note that
Aut(X )
CX ). Recall from [Hag07] that, after
X (which does nothing to the boundary
g ∈ Aut(X ) either xes a 0cube (elliptic) or
acts by isometries on
passing if necessary to the rst cubical subdivision of
and nothing serious to the contact graph), each
has a combinatorial geodesic axis (hyperbolic).
Let X be a CAT(0) cube complex with no innite set of pairwise-crossing hyperplanes. Let g ∈ Aut(X ). Then one of the following holds:
Corollary 2.7.
SIMPLICES AT INFINITY IN CAT(0) CUBE COMPLEXES
13
(1) g is elliptic;
(2) g stabilizes a clique in CX . In particular, if X is locally nite, then g N stabilizes a
hyperplane for some N ≥ 0;
(3) any axis of g bounds a combinatorial half-at, and there is a hgiorbit in CX of diameter
at most 3;
(4) g acts loxodromically on CX .
In the last case, g is a rank-one isometry of X .
Sketch.
Suppose
g
is hyperbolic and let
to a quasi-geodesic in
CX vertices
CX
(i.e.
A
be a combinatorial axis for
if the set of hyperplanes crossing
A
g
in
X.
If
A
projects
corresponds to a set of
spanning a quasi-line), then the last option holds. In this case,
g
must be rank-one
since, since half-ats have uniformly bounded projection to the contact graph. The rest of the
claim follows Theorem 2.4 by translating staircases or hyperplanes by powers of
g.
Remark 2.8. If you're into mapping class groups, and the Corollary looks like an unsatisfying
analogue of the Nielsen-Thurston classication, fear not, for soon we'll pass from the contact
graph to a more rened analogue of the curve graph which really will give a sense in which the
second two conclusions describe reducible elements.
Finally, we note that the Caprace-Sageev Irreducibility Criterion [CS11] combines with the
above classication to show that if
G acts geometrically and essentially on X , then the following
are equivalent:
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
X = A × B for unbounded subcomplexes A, B ;
CX is bounded;
CX = CA ? CB ;
∂4 X is connected;
∂4 X has no isolated 0simplex;
∂4 X = ∂4 A ? ∂4 B ;
G contains no rank-one isometry of X ;
G contains no loxodromic isometry of CX .
Later, we'll discuss how to prove this (in many interesting cases) without recourse to the
Irreducibility Criterion.
2.3.
Factor-systems and the factored contact graph. This part of the lectures will pre-
pare us for an alternative denition of
∂4 X
corresponding to a more sophisticated version of
projections of geodesic rays to quasi-trees, and the resulting compactication of
the following comes from [BHS14]. Given a convex subcomplex
projection map
gC : X → C :
given
x ∈ X (0) ,
C ⊆ X,
∂4 X .
Most of
there is a closest-point
there is a unique closest
0cube gC (x)
of
C
(exercise), and this extends in an obvious way to higher-dimensional cubes, so that they either
map cubically, or collapse onto faces and then map cubically.
The preceding discussion showed that a geodesic
or there is some hyperplane
H
so that
gN (H) (γ)
γ
either projects quasi-geodesically to
CX ,
is large. Presumably, there should be a way
to proceed inductively, and conclude that the distance along
γ
is coarsely the same as the sum
of a bunch of distances along hyperplanes, etc. (If you know about subsurface projections, this
should remind you of the Masur-Minsky distance formula...)
This plan looks doomed in the innite-dimensional case, and in fact the story is a bit more
complicated than the above, so we have to impose some additional restrictions on
Denition 2.9 (Factor system). A
(1)
X ∈F
(2) there
factor system F
is a set of convex subcomplexes so that:
H ∈ F for each combinatorial hyperplane H ;
(0) , we have |{F ∈ F : x ∈ F }| ≤ N ;
exists N so that for all x ∈ X
and
X.
SIMPLICES AT INFINITY IN CAT(0) CUBE COMPLEXES
(3) whenever
do
not
F, F 0 ∈ F,
either
gF (F 0 ) ∈ F
or
gF (F 0 )
14
is a single vertex, in which case we
include it (for a technical reason in the proof of Theorem 3.15; everything works
F).
then F = {X }. If X
just ne if we allow single points in
So, for example, if
F
consists of
X,
X
is a tree,
is the standard tiling of
R2
by
the hyperplanes (two parallelism classes of lines), and nothing
2cubes, then
else. If X is a
staircase, then it can't contain a factor-system (exercise; easy and important!).
Remark 2.10. The rst and second conditions imply that
particular nite dimension.
gF (F 0 )
X
has bounded degree, and in
The third condition can actually by relaxed somewhat, so that
need only be included if its diameter exceeds some predetermined threshold.
Remark 2.11. Lots of cube complexes have factor systems: if
so does any convex subcomplex of
X,
RAAGs have factor systems (exercises). In particular, if
then
G
X
has a factor system, then
and moreover universal covers of Salvetti complexes of
G is a virtually compact special group,
acts geometrically on a CAT(0) cube complex with a factor system. The exact scope of
the denition is the subject of Question
A0 !
F, F 0 ⊆ X are parallel if they intersect
CX . Given a factor-system F,
containing exactly one element F in each
Denition 2.12 (Parallel). Two convex subcomplexes
0
the same hyperplanes, i.e. if CF, CF are the same subgraph of
let
F
denote
F
mod parallelism (or, sometimes, a set
parallelism class
[F ]).
Denition 2.13 (Factored contact graph, projection). For each
F ∈ F, let FF be the set of
parallelism classes [E] ∈ F so that E is parallel to a proper subcomplex of F . Then dene F F to
be the set of parallelism classes in FF . To F (well, its parallelism class) we can now associate the
b which is obtained from CF by coning o CE ⊂ CF for each E ∈ FF .
factored contact graph CF
b
CF
There is a coarse projection πF : X → 2
: for each 0cube or open cube c of X , let πF (c) be
b
the clique in CF ⊂ CF whose vertices are the hyperplanes whose carriers contain gF (c).
b is (λ, λ)quasi-isometric to
There exists λ ≥ 1 (independent of X ) so that CF
a tree for each F ∈ F.
Theorem 2.14.
Remark 2.15. The factor system and the projections make
X
a
hierarchically hyperbolic space
(in fact, one of the main examples) in the sense of [BHS15].
Here's the most important consequence of the existence of factor systems.
Theorem 2.16 (Distance formula).
x, y ∈ X (0) ,
dX (x, y) C
For each suciently large s, there exists C so that for all
X
dCF
b (πF (x), πF (y)).
F ∈F,
dCF
b (πF (x),πF (y))≥s
Proof.
This uses some disc diagram considerations and induction on the multiplicity of the factor
system (passing from
F
to some
FF
lowers multiplicity). It's beyond the scope of these notes,
but most of the work lies in dening factor systems and factored contact graphs correctly; the
details are in [BHS14].
SIMPLICES AT INFINITY IN CAT(0) CUBE COMPLEXES
15
Problems on Part 2.
2.4.
C ⊆ X is a convex subcomplex, and x ∈ X (0) , then there is a unique closest 0cube
of C to x.
Let Γ be a nite graph and let SΓ be the Salvetti complex of the corresponding rightangled Artin group AΓ . Find at least two dierent (i.e. in principle dierent; maybe
eΓ that you can describe in terms of
they agree for some specic Γ) factor-systems on S
Γ and AΓ .
Let X be a cube complex with a factor system F and let Y ⊂ X be a convex subcomplex.
Prove that {F ∩ Y : F ∈ F} is a factor system.
(1) If
(2)
(3)
∗
(4) Prove Theorem 2.14. (Starred because tricky.)
(5) Give a nice combinatorial description of the convex hull of a geodesic ray.
∗
(6) Using the description of the convex hull of a geodesic ray, prove that, if
X
contains a
factor system, then it is fully visible. (Starred because interesting; this is sort of like
the bottleneck thing above, in the sense that you should start with a maximal simplex,
represent it by a ray (maximal simplices are visible), and then use the fact that factor
systems on
X
induce factor systems on convex subcomplexes to study the structure of
the convex hull of your ray...)
3. Part III: The compact simplicial boundary, rank-rigidity etc., and
Question
A
∂4 X using factor systems. We now assume X is a CAT(0) cube complex
with a factor system F. In particular, X has bounded degree and is thus nite dimensional (so
∂4 X is dened). We can and shall assume that F is the minimal factor system, i.e. it contains
X , every combinatorial hyperplane, and every other subcomplex needed to make it closed under
projection, but nothing else. As before, F is the set of parallelism classes, which indexes the set
b .
of factored contact graphs CF
3.1.
Redening
In the previous exercises, it was proved that:
Proposition 3.1.
If X has a factor system, then X is fully visible.
We're going to apply the distance formula to re-imagine the simplicial boundary in a way
that explicitly involves the factored contact graphs. The goal is to build a compact version of
the simplicial boundary. As an application, we'll give a new proof of rank-rigidity, under the
assumption that factor-systems exist.
Rank-rigidity, its consequences, and related things are
awesome, so this will complete the mission to motivate Question
A!
The following discussion appears in a dierent form in the forthcoming paper [DHS15], but
the one below is more streamlined; things are easier the second time one does them.
F, F 0 ∈ F. We
F, F 0 ,→ X extends to
Denition 3.2 (Nesting, orthogonality, complement). Let
write
F @ F0
if
F
is
0
0
parallel to a subcomplex of F . We write F ⊥ F if
a convex embedding
0
F × F → X . One can check that if P is a set of pairwise-parallel elements of F, then there is a
F × E → X so that each element of P has the
e ∈ E . If P is an entire parallelism class, represented by some F ∈ F,
E = EF is the complement of F .
convex subcomplex
form
F × {e}
E
and a convex embedding
for some
then the complex
Denition 3.3 (γ relevant). Given a combinatorial geodesic ray
be the subgraph of
γ relevant
if
πF (γ)
b
CF
spanned by the hyperplanes of
F
γ
X and F ∈ F, let πF (γ)
γ . We say that F ∈ F is
of
that cross
is unbounded.
b , and hence picks
Let F ∈ R(γ). Then πF (γ) is a quasigeodesic ray in CF
F
b (the Gromov boundary). Moreover, R(γ) 6= ∅.
out a unique point pγ ∈ ∂ CF
Proposition 3.4.
SIMPLICES AT INFINITY IN CAT(0) CUBE COMPLEXES
Proof.
16
Distance formula!
Let γ be a combinatorial geodesic ray. Then the set R(γ) of relevant elements
of F is pairwise-orthogonal and hence contains at most n + 1 elements, where γ represents an
nsimplex of ∂4 X .
Proposition 3.5.
Proof.
Let
v = [v0 , . . . , vn ] be the simplex of ∂4 X represented by γ .
n. You showed in an exercise that, since X has a
by induction on
We will prove the claim
factor system, it is fully
visible. Moreover, it is clear that the claim depends only on the asymptoty class of
may modify
γ
in this way if needed. Hence, as shown in Section 3 of [Hag13], for each
choose a geodesic ray
P =
Q
i γi → X
γi
representing
γ , so we
i we can
vi so that there is a combinatorial isometric embedding
γ as a ray emanating from the basepoint and crossing
whose image contains
every hyperplane crossing
Principle I: Let
H(γ)
P.
be the set of hyperplanes crossing
γ.
H(γ), F 0
If
F, F 0 ∈ R(γ),
F ∩
F = F0 =
then
∩ H(γ) are both innite. If these sets have innite intersection, then either
gF (F 0 ) = gF 0 (F ), or F 00 = gF (F 0 ) is a proper subcomplex of F crossed by innitely many
b is a subgraph consisting of vertices all
hyperplanes crossing γ . So the projection of γ to CF
0
b
b
adjacent to the cone-point over CF 00 ⊂ CF , contradicting that F ∈ R(γ). Thus if F, F ∈ R(γ)
0
are distinct, then each of F, F can cross only nitely many hyperplanes crossing γ .
The base case n = 0: If F, F 0 ∈ R(γ) and n = 0, then H(γ) is a minimal boundary
0
set. In particular, if innitely many elements of H(γ) cross F , and innitely many cross F ,
0
0
then innitely many cross F and F . So Principle I tells us F = F . In particular, R(γ) is a
pairwise-orthogonal set with ≤ n + 1 elements.
Qn−1
0
00
The inductive step: Write P 0 =
i=0 γi , so that P = P × γn . Choose a diagonal ray γ
0
00
00
in P crossing every hyperplane, so that P contains γ × γn and so that R(γ) = R(γn ) t R(γ )
00
(since R(γ) just depends on the set of hyperplanes crossing γ ). By induction, R(γ ) is a pairwiseorthogonal set with ≤ n elements, and R(γn ) is a singleton since vn is a 0simplex. Let Fn be
00 be the product of the elements of R(γ 00 ), which is a
the unique element of R(γn ) and let F
convex subcomplex.
H be a hyperplane crossing γ 00 (hence F 00 ). Then H crosses each hyperplane crossing γn ,
whence gH (Fn ) @ Fn contains the projection of γ , so gH (Fn ) = Fn , i.e. Fn @ H , by Principle I.
00
0
00
This holds for any H crossing γ . Similarly, for any V crossing γn and any F ∈ R(γ ), we have
0
F @ V . Another application of Principle I now shows that every hyperplane crossing F 00 crosses
every hyperplane crossing Fn . Applying Proposition 2.5 of [CS11] completes the proof.
Let
Remark 3.6 (Nielsen-Thurston classication). This is a digression, but we can now give the
X is a CAT(0) cube complex with a factor system
g ∈ Aut(X ), then on of the following holds:
• g is elliptic;
• g is reducible axial : there exists N > 0 and F ∈ F − {X } so that g N stabilizes F and
b ;
acts loxodromically on CF
b .
• g is irreducible axial : g acts loxodromically on CX
promised Nielsen-Thurston classication. If
F,
and
The proof is an exercise and the digression is over.
R = {F0 , . . . , Fn } of pairwiseb
orthogonal elements of F, and each choice (p0 , . . . , pn ) ∈
nsimplex, which
i ∂ CFi , we put an P
Pn
we regard as the set of formal sums
a
p
,
where
each
a
∈
[0,
1]
and
i
i=0 i i
i ai = 1. (This
makes the face relation obvious, and clearly ∆X is a ag complex.)
We now dene a simplicial complex
∆X
as follows: for each set
Q
Proposition 3.7.
Sketch.
∆X is isomorphic to ∂4 X .
There are a few steps. A detailed proof is carried out in Section 10 of [DHS15].
SIMPLICES AT INFINITY IN CAT(0) CUBE COMPLEXES
17
Check that X is fully visible: This was an exercise!
Represent boundary points with combinatorial rays: For each
let
γ
F.
From the denition of a minimal boundary set, it follows that
be a combinatorial geodesic ray representing
v
and let
R(γ)
be the
R(γ)
0simplex v
γ relevant
of
∂4 X ,
elements of
has a unique element,
F = F (γ). Indeed, if F, F 0 ∈ R(γ), then they are orthogonal, so innitely many hyperplanes
0
0
crossing γ cross F (but not F ) and innitely many cross F (but not F ). These innite sets
contain sub-boundary sets of H(γ), contradicting that dim v = 0. Let πF (γ) be the (uniformly
b , representing a point pv ∈ ∂ CF
b .
quasigeodesic) projection of γ to CF
The isomorphism I : Dene I : ∂4 X → ∆X as follows. For each 0simplex v , let I(v) = pv ,
0
where p(v) is the 0simplex of ∆X dened above. Let v, v be adjacent 0simplices of ∂4 X .
say
Then, since they're visible, we have (up to changing the basepoints of the rays) a cubical
γ × γ 0 → X , where γ, γ 0 represent v, v 0 , as shown in [Hag13]. Hence there
00
0
0
is a diagonal geodesic ray γ representing the simplex v ? v , lying in γ × γ , and having the
0
0
same projection as γ (resp. γ ) on F (γ) (resp. F (γ )) and its factored contact graph. Thus
F (γ), F (γ 0 ) are in R(γ 00 ) and thus orthogonal by Proposition 3.5. Thus there is a 1simplex
[pv , pv0 ] in ∆X , and we let I([v, v 0 ]) = [pv , pv0 ]. Higher simplices can be handled similarly, or
you can use that ∆X and ∂4 X are both ag complexes, or you can induct, or whatever.
isometric embedding
We've now understood points in the simplicial boundary in terms of the Gromov boundaries
of the factored contact graphs. Since the factored contact graphs are (locally innite, in general)
quasi-trees, their boundaries have interesting but understandable topology, which we're going
to exploit to build a sort of contact graph cone topology on
∂4 X .
There's an extra wrinkle
introduced by the orthogonality relations/the many non-asymptotic rays that in general represent the same simplex, i.e. are associated to the same collection of rays in factored contact
graphs, namely that dierent rays can traverse their shadows on the factored contact graphs at
dierent speeds (or, if you prefer the old-school simplicial boundary interpretation, they can
punch through their dierent minimal boundary sets in dierent orders). This is part of the
reason for the horrendousness of the denition of the topology that we now dene.
Let X be a CAT(0) cube complex with a factor system F.
Then there is a space ∂f X with the following properties:
Theorem 3.8 (The HHS boundary).
(1)
(2)
(3)
(4)
(5)
∂f X ∪ X is a compact, Hausdor, separable space;
X is dense in ∂f X ∪ X ;
there is a continuous bijection ∂4 X → ∂f X which is an embedding on each simplex;
b → ∂f X for each F ∈ F;
there is an embedding ∂ CF
b is nonempty (i.e. CX
b is unbounded), and there is a group acting geometrically
if ∂ CX
b
on X , then ∂ CX is dense in ∂f X .
Remark 3.9. It follows easily from the denitions that
b
CX
is quasi-isometric to
CX
the
former is obtained by coning o bounded subgraphs of the latter so the latter part of the
theorem is related to the question of when the contact graph is bounded.
Remark 3.10. In fact, the boundary mentioned in the theorem is a special case of the
ary of a hierarchically hyperbolic space (HHS)
in [DHS15].
bound-
introduced by Durham, the author, and Sisto
This same construction also yields a bordication of the mapping class group,
a new bordication of Teichmüller space, universal covers of non-geometric
3manifolds,
etc.
From the HHS point of view, the cubical case is a particularly nice test case, both because the
hyperbolic spaces out of whose Gromov boundaries we're building the boundary are particularly
simple (quasi-trees) and because there's an alternative interpretation of the boundary (well, not
topologically, but more than just as a set), namely the original denition of
∂4 X .
SIMPLICES AT INFINITY IN CAT(0) CUBE COMPLEXES
18
Remark 3.11 (Bounded Geodesic Image). In one spot in the following discussion of Theorem 3.8, we need the cubical Bounded Geodesic Image theorem which we now briey discuss.
Let
H, F ∈ F
and suppose
F @ H.
For concreteness, since we'll be working at the level of the
factored contact graphs, we can just assume that
x∈X
F ( H.
Now, for any
b ,
t ∈ CH
we can nd
b by considering πF (x). This
t, and then send it to CF
b
b
gives a (coarsely surjective) coarse map CH → CF (there's a map the other way, too, which
b to the cone-point in CH
b over CF
b , or use the
is less interesting: just send everything in CF
that projects uniformly close to
inclusion, or whatever).
Now, suppose that
b .
CF
γ
is a (quasi)geodesic in
0
Then for any points t, t
∈ γ,
b
CH
that does not pass through the subgraph
the corresponding points
x, x0 ∈ X
can be chosen so that
each hyperplane separating them (or, depending on the exact situation, all but uniformly many
hyperplanes separating them) fails to cross
bounded. It follows that if
then there is a point
t∈γ
γ
F.
Thus the image of
γ
under
b → CF
b
CH
is uniformly
is a (quasi)geodesic ray originating near the cone-point over
so that
b → CF
b
CH
is coarsely constant on
γ([t, ∞)).
b ,
CF
This technical
point plays a fairly major role in this business, and is a factor system analogue of the bounded
geodesic image theorem in the mapping class group [MM00], but in the present text it only
plays a minor technical role, in the denition of the topology on
∂f X
below.
Discussion of the proof of Theorem 3.8. As a set, ∂f X is just ∆X . The most illuminating part
∆X yielding the boundary ∂f X , which is explained below.
Proposition 3.7 provides the bijection ∂4 X → ∂f X ; it's just the identity (one must check it's
b ,→ ∂f X (which
an embedding on simplices). The denition of ∆X provides the inclusions ∂ CF
one must check are embeddings). The denition of the topology will make it obvious that ∂f X
is the denition of the topology on
is Hausdor and separable.
The details of the proof are in [DHS15] and are more or less technical applications of the
denitions, although the proof of compactness is quite involved. In [DHS15], we work in the
context of hierarchically hyperbolic spaces. As explained in [BHS14], the factor system
the projections prove that
X
F
and
is an HHS, so that the arguments in [DHS15] apply. Section 10
of [DHS15] also discusses the relationship between
∂4 X
and
∂f X ,
in a slightly dierent way.
We now discuss how to dene the topology. Specically, we'll build a neighborhood basis for
X ∪ ∂f X consisting of open balls in X together with sets that we now dene.
P
• Let p = F ∈R aF pF , where R is a set of parallelism classes in F with pairwise-orthogonal
b . Let > 0, and let OF be
representatives, and each aF ∈ (0, 1), and each pF ∈ ∂ CF
b
b
a neighborhood of pF in CF ∪ ∂ CF (with the cone topology). We will associate a set
N,{OF } (p) ⊂ X ∪ ∆X to this data, and the set of all such sets will be our neighborhood
a topology on
basis.
•
Let's rst decide what it means for an interior point
close to
p,
x ∈ X
to be in
N,{OF } .
To be
it seems reasonable, in view of the distance formula, that we should require
x to project close to p in each relevant factored contact graph, i.e.
Q πF (x) ∈ OF for each
F ∈ R. That's not enough, though: there's a whole orthant in F ∈R F with the given
property; we also need to account for the slope of the ray representing p. Accordingly,
xing once and for all a basepoint x0 ∈ X , we require that
aF
dCF
b (πF (x0 ), πF (x)) a 0 − d(π 0 (x0 ), π 0 (x)) < F
F
F
for
x
F, F 0 ∈ R.
Finally, we don't want some
has some large
H coordinate
H ∈ F,
orthogonal to every
F ∈ R,
so that
dragging it out of the orthant corresponding to the
pF .
SIMPLICES AT INFINITY IN CAT(0) CUBE COMPLEXES
Therefore, we require that for all such
H
and all
19
F ∈ R,
dCH
b (πH (x0 ), πH (x))
< .
dCF
b (πF (x0 ), πF (x))
interior part of our basic set N,{OF } .
b
• Let q =
H∈R0 bH qH be a boundary point, with qH ∈ ∂ CH and bH ∈ (0, 1). To
test whether q ∈ N,{OF } (p), we have to test whether q lies in approximately the same
b F ∈ R, and that q does not make much progress away
direction as p as measured in CF,
from p elsewhere. To formalize this, we have to consider two types of q :
Non-generic: Here, either R ∩ R0 6= ∅, or there exists H ∈ R0 which is orthogonal
00 = R ∩ R0 . In order to include q in our
to each F ∈ R. In this case, let R
This denes the
P
neighborhood, we require that:
00
∗ qP
H ∈ OH for all H ∈ R ;
∗
H∈R0 −R00 bH < (i.e. most of
with p);
∗ |aH − bH | < for all H ∈ R00 .
the action is in directions
q
has in common
Generic: The nal case is that where R ∩ R0 = ∅ and, moreover, each
not orthogonal to some HF ∈ R0 . First we require that
X
F ∈R
is
bH < H
H ∈ R0 that are orthogonal to every F ∈ R.
b for each F ∈ R and require
Next, we are going to dene a projection πF (p) ∈ CF
that q project close to the pdirection in the prelevant factored contact graphs,
, where the sum is taken over the
with the right contribution from each one, i.e.:
∗ πF (q) ∈ OF for each F ∈ R;
∗ for all F, F 0 ∈ R,
dCF
(πF (x0 ), πF (q))
aF b
d 0 (π 0 (x0 ), π 0 (q)) − a 0 < .
F
F
F
b
CF
πF ? Well, by denition we have some H ∈ R0 so that H is neither
orthogonal to F nor equal to F . One can check that this provides a uniformly
b obtained by closest-point projecting H to F and then
bounded subset ρ of CF
b
projecting to CF unless F @ H . So, in this case, we let πF (q) = ρ; this is forced
b we're
on us: we'd better be projecting the boundary of H to the same place in CF
projecting the rest of H !
b from the
Otherwise, F @ H . In this case, let γH be a quasigeodesic ray in CH
b
b
b
cone-point over CF ⊂ CH to qH ∈ ∂ CH . Bounded Geodesic Image provides a rst
point t ∈ γH so that γH is coarsely constant after t. Choose x ∈ X so that πH (x)
coarsely coincides with t, then let πF (q) = πF (x).
completes the description of the topology.
How to dene
This
Example 3.12 (Examples). Two simple examples:
(1) If
X
is a hyperbolic CAT(0) cube complex with a proper cocompact group action, then
Agol's virtual specialness theorem combines with results of [BHS14] to show that
a factor system. As one would hope and expect,
∂f X
X
has
turns out to be homeomorphic to
the Gromov boundary in this case, as shown in [DHS15].
(2) If
to
X decomposes as a product of
∂4 X , which (recall) is a join.
unbounded subcomplexes, then
∂f X
is homeomorphic
SIMPLICES AT INFINITY IN CAT(0) CUBE COMPLEXES
3.2.
20
Stationary measures on the boundary, rank-rigidity, and the like. We'll focus
on uses of
∂f X
that involve the next few lemmas.
more general context of
Everything in this section works in the
hierarchically hyperbolic spaces
(see [BHS14, BHS15, DHS15]), but the
cubical case is more concrete but does not hide any of the essential ideas. Also, one can relax
cocompactness in various ways. (Besides, these are cubical lectures.)
In this section, we work in the setting of a group
G
a CAT(0) cube complex equipped with a factor-system
measure
µ
whose support generates
Using compactness of
∂f X ,
G
(use that
G
one can construct a
acting geometrically on
F.
Equip
G
X,
which is
with a Borel probability
is nitely-generated if you want).
µstationary
measure
ν
on
X ∪ ∂f X ,
i.e. a
Borel probability measure such that
ν(E) =
X
µ(g)ν(g −1 E)
g∈G
E ⊆ X ∪ ∂f X .
Consider the action of G on the set F induced by the action on F and the fact that G preserves
parallelism. Note that X is xed.
for all
ν measurable
subsets
b , then ν is
If G has no nite orbit in F − {[X ]} and no nite orbit in ∂ CX
b ⊂ ∂f X .
supported on ∂ CX
b ) < ∞, there exists F ∈ F − {X } and G0 ≤f.i. G so that gF is parallel to F
Hence if diam(CX
0
for all g ∈ G .
Lemma 3.13.
Proof.
The second assertion follows from the rst since
ν
can't be supported on
∅,
so it remains
to prove the rst assertion.
D be the set of nite subsets of F, so that G has no nite orbit in D other than G · {[X ]}
b and we're done.
and G · ∅. We can assume there exists F ∈ F − {X }; otherwise ∂f X = ∂ CX
We'll dene a Gequivariant map O : X ∪ ∂f X → D which is measurable when D is endowed
−1 (A)) for all A ⊆ D . We will also check
with the probability measure ν̃ given by ν̃(A) = ν(O
b is measurable and O(E) contains no nite Gorbit. It follows from
that E = X ∪ ∂f X − ∂ CX
e.g. [KM96, Lemma 2.2.2],[Woe89],[Bal89],[Hor14] that ν(E) = 0, and we're done.
So: if p ∈ ∂f X is represented by some ray γ , let R(γ) be the set of relevant factor-system
b .
elements, and let O(p) = R(γ). This is Gequivariant, and O(p) = {[X ]} if and only if p ∈ ∂ CX
To dene the map on X , let C ⊂ X be a subset which contains exactly one open cube of X in
each Gorbit, and exactly one 0cube in each Gorbit. Since G acts geometrically, C is a nite
union of Borel subsets of X ∪ ∂f X , so C is measurable. Fix a set A ∈ D − {∅, [X ]}, which exists
because we can use {F }. Let O(x) = A for each x ∈ C , and extend Gequivariantly to all of
GC = X . From the denition, one can easily check that O is Borel (use that G is countable).
Since X is an open subset of X ∪ ∂f X , it follows from Lemma 3.14 that E is measurable.
Finally, if p ∈ X , then G · O(p) = G · A, which is innite. If p ∈ ∂f − ∂X , then GO(p) must
be innite, since otherwise it would be a nite union of subsets of F − {[X ]}, which cannot be
Ginvariant by hypothesis.
Let
Lemma 3.14. For any nite subset R of F, the set of p ∈ ∂f X whose set of relevant factored
contact graphs is R is a Borel set.
Proof.
12 lines in [DHS15], but not fun lines) through the denition
exercise.
This is a horrible chase (it's
of the topology on the boundary;
We can now reprove a special case of the Caprace-Sageev rank-rigidity theorem [CS11], as a
test of our tool.
Let X be a CAT(0) cube complex with
a factor system F. Suppose that some group G acts properly and cocompactly on X . Then one
Theorem 3.15 (Rank-rigidity assuming factor systems).
SIMPLICES AT INFINITY IN CAT(0) CUBE COMPLEXES
21
of the following holds provided G acts essentially on X in the sense that every halfspace contains
Gorbit points arbitrarily far from the associated hyperplane:
(1) CX is unbounded, and G contains a rank-one isometry of X acting loxodromically on
CX .
(2) CX is bounded, and X splits as the product of two unbounded convex subcomplexes.
The second conclusion holds if and only if ∂4 X decomposes as a nontrivial simplicial join.
Remark 3.16. Caprace-Sageev require a proper, cocompact, essential action
action with no xed point at innity.
or
an essential
Our proof actually works in the second context, too
(see [DHS15]), but in either case, we need a factor-system (while [CS11] does not). A positive
answer to Question
A
would allow us to drop that hypothesis in the cocompact setting. More
to the point (since rank-rigidity already has a nice proof ), it would hopefully allow one to solve
other problems using these techniques, in the general setting of cocompactly cubulated groups.
Proof of Theorem 3.15.
b = ∅ (since CX
b and CX are
∂ CX
q.i.). Lemma 3.13 provides F ∈ F − {X } so that (up to nite index) G preserves the parallelism
⊥
⊥ is bounded, then essentiality tells us
class of F . Essentiality tells us that X = F × F . If F
it's trivial, so X = F , a contradiction. Otherwise, F is bounded, so is trivial by essentiality,
First suppose that
CX
is bounded, i.e.
which we disallowed. Hence the second conclusion holds.
Hence suppose that
CX
is unbounded (i.e.
b
CX
is unbounded). There are two ways to proceed.
Method I: use Caprace-Sageev double-skewering: Let
H, H 0
be hyperplanes corre-
CX . By the Double-Skewering Lemma [CS11],
0
0
there exists a hyperbolic element g ∈ G so that H separates H and gH (i.e. H, H cut the axis
of g ). It is easy to check (Exercise 3.3.(4)) that hgi has an unbounded orbit in CX , so our above
discussion about projecting geodesic rays to the contact graph, applied to an axis of g , shows
that g is loxodromic on CX and in particular rank-one.
sponding to vertices at distance at least
10
in
Method II: use acylindricity: In [BHS14], it is shown that
b .
CX
A result of Osin [Osi15] combines with the fact that
b ,
contains a loxodromic isometry of CX
have diameter-≤ 3 projection to CX ).
b
CX
G
acts
acylindrically
is unbounded to show that
on
G
which is necessarily rank-one as above (since half-ats
The key here (from the point of view of the boundary; we also used other serious tools) was
Lemma 3.13, which should provide a template for proving various similar results (in the more
general HHS situation, one can use it to prove things like the Tits alternative or a generalization
of the Handel-Mosher omnibus subgroup theorem for mapping class groups [DHS15]; hopefully
in the more restricted setting of cube complexes with factor systems, one can do even more.
We've focused heavily on the case where
X
has a factor system; indeed, we've mainly left
aside things that one can do with the simplicial boundary in general (see e.g. [BH]). Now, the
denition of a factor system is not particularly natural-seeming, even if it is what one is led to
when trying to write down a Masur-Minsky-style distance formula for right-angled Artin groups
(factor systems are not the rst attempt at the latter [KK14]). It would be nice if, at least in
the case of cube complexes with proper, cocompact group actions, one had access to the factor
∂f X , and the methods we've illustrated above, without having
F. This is what a positive answer to Question A would enable.
system technology, the boundary
to hypothesize the existence of
SIMPLICES AT INFINITY IN CAT(0) CUBE COMPLEXES
3.3.
22
Final problems. We nish with a list of problems related to Part 3, and some other
interesting problems related to (or possibly approachable using) the boundary. The latter list is
biased toward things I think are very interesting but don't know how to do, or total speculation.
Problems on
∂f X :
(1) Prove the main property of orthogonal complements:
a maximal convex subcomplex
F⊥
so that
F → X
for each
F ∈ F,
there exists
extends to a convex embedding
F × F ⊥ → X with the property that each parallel copy of F is the image of F × {e} for
⊥
some vertex e ∈ F . (Hint: take the convex hull of the union of all parallel copies of
F , and then nd two classes of hyperplanes, forming a join in the contact graph, from
which you can read o the product decomposition.)
(2) Prove the Nielsen-Thurston classication outlined in Remark 3.6.
(3) Prove Lemma 3.14.
(4) Check that if
H, H 0
are hyperplanes at large contact-graph distance, both cutting an
A of a hyperbolic isometry g of X , then A has unbounded projection to CX .
2
Let R2 denote the standard tiling of E by 2cubes, and let X be the obvious tree of
2
2
R2 s on which Z ∗ Z acts geometrically. How is CX related to the Bass-Serre tree?
Describe ∂f X .
axis
(5)
∗
(6) What else can you prove about cube complexes with factor systems, and groups acting
∗
(7) For interesting non-hyperbolic cube complexes
on them, using Lemma 3.13 or related ideas? (E.g. Tits alternative...)
X
with factor systems, arising in nature
(e.g. from RAAGs, Coxeter groups, etc.), describe
∂f X
in a satisfying topological way.
(In other words, prove theorems along the lines of various theorems describing Gromov
boundaries of hyperbolic groups.)
Problems on relative hyperbolicity, thickness, divergence, quasi-isometries: In the
following problems,
X
is a CAT(0) cube complex on which the group
G acts geometrically (and,
say, essentially, although this is only necessary for some parts).
∗
(8) In [BH], the existence of nontrivial relatively hyperbolic structures on
in terms of the structure of
to subgroups
{Pi }
if
∂4 X
∂4 X
and the action of
consists of a
G.
Ginvariant
Roughly,
G
G is characterized
is hyperbolic relative
collection of isolated
0simplices,
together with a bunch of subcomplexes which are the simplicial boundaries of convex
Pi
∂4 X ,
subcomplexes stabilized by the various
and their conjugates.
completely in terms of the action on
although it is ddly.
This can be phrased
In [BDM09], Behrstock-Drutu-Mosher introduced the notion of a
thick space.
Thick-
ness is an obstruction to the existence of a nontrivial relatively hyperbolic structure, and
is enjoyed by lots of the usual suspects, like one-ended RAAGs, mapping class groups,
Out(Fn ),
etc.
X
is
thick of order 0
if it's a product with unbounded factors [Hag13]
(there is a more general denition for general metric spaces: no cutpoints in any as-
n if there is a collection {Ui }
n − 1, so that any two points in the main
space are connected through a chain of elements of {Ui } so that successive elements have
ymptotic cone).
Inductively, a space is thick of order
of undistorted subspaces, all thick of order
coarsely connected, unbounded intersection. In [BH], we characterised thickness of order
1 for G completely in terms of ∂4 X and the action of G thereon; roughly, it corresponds
to the existence of a connected Ginvariant subcomplex, but with the overall boundary
being disconnected.
Must G have a relatively hyperbolic structure whose peripheral subgroups
have no nontrivial RH structure? Whose peripheral subgroups are thick?
Maybe studying
∂4 X
is useful here. Note that this is true for Coxeter groups [BHS],
which are cubulated [NR03], though not always cocompactly. This might involve
gen-
eralizing the above result on thickness of order 1: can one identify higher-order
SIMPLICES AT INFINITY IN CAT(0) CUBE COMPLEXES
thickness of
G
from
∂4 X ? Note
n [BH].
that for any
k, n
there is a
23
k dimensional X
so that
G
is thick of order exactly
Thickness is also related to the divergence function of
its divergence is polynomial of order at most
n + 1,
G:
if
G is thick of order n, then
see [BD14]. In general, it is hard
If G is thick of order n, is the divergence
X exactly polynomial of order n + 1? Must the divergence function of
to prove lower bounds on divergence.
function of
X
be either polynomial or exponential? (A positive answer would follow from a positive
answer to the above question on minimal RH structures with thick peripherals, and a
positive answer to the question about divergence of thick cubical groups.)
Other problems:
∗
(9)
Graph colouring and Question
A.
Let
C] X
be the intersection graph of the hyper-
crossing graph ). Suppose that h : C] X → Γ is a graph homomorphism
v, w ∈ C] X (0) are adjacent whenever h(v), h(w) are adjacent, and
suppose that Γ is nite. (Call h a Γcolouring of X .)
Note that if X admits a Γcolouring for some nite Γ, then C] X has nite chromatic
number. In fact, this corresponds to an isometric embedding of X in a nite product
planes in
X
(the
with the property that
of trees, and this is not always possible even when you remove obvious restrictions by
bounding the degree of
0cubes
in
X
(see [CH13]). However, if
X
has a factor system
and a geometric group action and satises one additional mild technical hypothesis,
then
X
admits a very particular type of
Γcolouring
corresponding to an
equivariant
embedding in a nite product of trees [BHS14].
X admits a Γcolouring h : C] X → Γ. Let SΓ be the Salvetti complex
associated to Γ, with 1cubes labelled by vertices of Γ and, more generally, ncubes
labelled by cliques in Γ. Dene a map q : X → SΓ as follows. First, q(x) is the unique 0
cube of SΓ for each 0cube x ∈ X . For each 1cube e of X , let H be the dual hyperplane,
and send e to the 1cube of SΓ labelled by h(H). Extend in the obvious way to higher
cubes. This map is locally injective because no two 1cubes with a common endpoint
in X are dual to the same hyperplane. It's a cubical map since h is a homomorphism.
It's a local isometry since any cube in SΓ whose corner is in the image pulls back to X
eΓ . In other
by the second condition on h. Hence q lifts to a convex embedding X → S
words, Question A can be rephrased as: if X admits a proper, cocompact group action,
does it admit a Γcolouring for some nite Γ. Does this help?
Restricted triple space. Let X be any CAT(0) cube complex for which ∂4 X is
dened (i.e. no factor system assumptions). A triple of distinct points p, q, r ∈ ∂4 X
is visible if there do not exist distinct a, b ∈ {p, q, r} so that a ∈ v, b ∈ w where v, w
are simplices with nonempty intersection. Let V be the set of visible triples. Using
Proposition 1.14 and the relationship between the Tits boundary and ∂4 X , along with
(1) is a median graph, one should be able to construct a (coarse) map
the fact that X
V → X roughly as follows: given a visible triple p, q, r, form an ideal triangle of biinnite combinatorial geodesics whose endpoints are p, q, r . Then, for each t ≥ 0, travel
distance t in the positive and negative directions on each of these geodesics, to give you
So, suppose
∗
(10)
three pairs of (uniformly close) points. Take the median (coarsely) and show that this
stabilizes as
t → ∞.
Can
V
be topologized so that this map is useful? Is there some
notion of a uniform convergence action related to this construction? What if
factor system and you're using triples in
∗
(11)
∂f X
X
has a
instead?
Simplicial-ish boundary of a median space? Dene an analogue of the simplicial
boundary of a measured wallspace, or for the dual median space (see [CDH10]). What
can you do with this object?
SIMPLICES AT INFINITY IN CAT(0) CUBE COMPLEXES
∗
(12) What topological spaces can arise as
∂f X
where
X
24
is a CAT(0) cube complex with a
factor system?
∗
(13) If
∗
(14) If
X
admits two factor systems, must the associated compactied simplicial boundaries
be homeomorphic? (I guess yes, and maybe this is not so hard, in view of Theorem 3.8.)
G
and
acts properly and cocompactly on CAT(0) cube complexes
∂4 Y
be isomorphic simplicial complexes?
Homeomorphic?
X
and
Y,
must
∂4 X
(Croke-Kleiner-type
∂4 X has no idea which CAT(0) metric you're using.
G is virtually Zn , then ∂4 X has to be the hyperoctahedron
think the isomorphism ∂4 X → ∂4 Y has to be equivariant in
problems [CK00] don't arise because
It is shown in [Hag14a] that if
you expect, but I don't
general, when there is an interesting point group. The point of [Hag14a] was to make
an argument that generalizes to more interesting cubical groups, but it's not clear that
it's the best strategy to follow...)
∗
(15) Let
G
act geometrically on
subgroup
A≤G
X.
It is shown in [WW15] that each virtually abelian
which is highest, in the sense that no nite-index subgroup of
A
is
contained in a higher-rank abelian subgroup, stabilizes a convex subcomplex of the form
Q
i Ci where each
Ci is the convex hull of a combinatorial line (and lies at nite Hausdor
A stabilizes a hyperoctahedron in the boundary, namely
distance from that line). Hence
∂4
Q
i Ci . Can this be used to generalize the main result of [Hag14a] by proving that
certain groups cannot admit a cocompact cubulation? Coxeter groups come to mind,
here. (Something like: nd a virtually abelian subgroup that doesn't act the right way
on the required hyperoctahedron (like a
results of [WW15] and [Hag14a].)
(3, 3, 3)
triangle subgroup), then combine the
SIMPLICES AT INFINITY IN CAT(0) CUBE COMPLEXES
25
References
[Bal89]
[BD14]
[BDM09]
[BH]
[BHS]
[BHS14]
[BHS15]
[CDH10]
[CH13]
[CK00]
[CS11]
[DHS15]
[Hag07]
[Hag13]
[Hag14a]
[Hag14b]
[Hor14]
[KK14]
[KM96]
[MM99]
[MM00]
[NR03]
[Osi15]
[Woe89]
[WW15]
Werner Ballmann. On the dirichlet problem at innity for manifolds of nonpositive curvature. In Forum
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Jason Behrstock and Cornelia Druµu. Divergence, thick groups, and short conjugators. Illinois Journal
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Jason Behrstock, Cornelia Druµu, and Lee Mosher. Thick metric spaces, relative hyperbolicity, and
quasi-isometric rigidity. Mathematische Annalen, 344(3):543595, 2009.
Jason Behrstock and Mark F. Hagen. Cubulated groups: thickness, relative hyperbolicity, and simplicial boundaries. To appear in Groups, Geometry, and Dynamics.
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DPMMS, University of Cambridge, Cambridge, UK
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