Coning-off CAT(0) cube complexes
Anthony Genevois
March 22, 2016

arXiv:1603.06513v1 [] 21 Mar 2016

Abstract
In this paper, we study the geometry of cone-offs of CAT(0) cube complexes over
a family of combinatorially convex subcomplexes, with an emphasis on their Gromovhyperbolicity. A first application gives a direct cubical proof of the characterization of the
(strong) relative hyperbolicity of right-angled Coxeter groups, which is a particular case of
a result due to Behrstock, Caprace and Hagen. A second application gives the acylindrical
hyperbolicity of C 0 (1/4) − T (4) small cancellation quotients of free products.

Contents
1 Introduction

2

2 Preliminaries

4

3 Hyperbolicity with respect to the `1 and `2 metrics

11

4 Killing flat rectangles

14

5 Relative hyperbolicity
5.1 Weak relative hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Strong relative hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Application to right-angled Coxeter groups . . . . . . . . . . . . . . . .

16
16
18
20

6 Hyperbolicity with respect to the `∞ metric

26

7 Acylindrical actions

28

8 Acylindrical hyperbolicity of infinitely-presented groups
8.1 Small cancellation polygonal complexes . . . . . . . . . . .
8.2 A fixed point theorem . . . . . . . . . . . . . . . . . . . . .
8.3 Application to small cancellation quotients . . . . . . . . .
8.4 A note on universal actions . . . . . . . . . . . . . . . . . .
References

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

29
29
31
34
40
43

1

1

Introduction

A fruitful method to study a given group is to make it act on some space which
is “negatively-curved” in some sense. Following Gromov, a possibility is to define δhyperbolic geodesic spaces by requiring that, in any geodesic triangle, any side is included
into the δ-neighborhood of the union of the two other sides. Often it happens that a
group G admits a natural action on some geodesic space X, which is generally not
hyperbolic however. A well-known method to produce an action on a hyperbolic space
is to cone-off this space: loosely speaking, we construct a new geodesic space Y from X
by gluing cones over subspaces of X, in order to “kill” the non-hyperbolic subspaces of
X and to make Y hyperbolic; thus, if this collection of subspaces is G-equivariant, the
action G y X naturally induces a new action G y Y . In this paper, for convenience
we will use two different definitions of a cone-off; nevertheless, the spaces we obtain are
quasi-isometric:
Definition 1.1. Let X be a CW complex and Q a collection of subcomplexes. The
cone-off of X over Q is the graph obtained from X (1) by adding an edge between two
vertices whenever they both belong to a common subcomplex of Q.
Definition 1.2. Let X be a CAT(0) cube complex and Q a collection of subcomplexes.
The usual cone-off of X over Q is the graph obtained from X (1) by adding a vertex for
each subcomplex Q ∈ Q and linking it by an edge Q to each vertex belonging to Q.
In this article, we focus on the class of CAT(0) cube complexes: roughly speaking, a
CAT(0) cube complex is a geodesic space constructed by gluing cubes together, in such
a way that the geodesic triangles turn out to be thiner than the Euclidean triangles.
Essentially, we determine precisely when a CAT(0) cube complex is hyperbolic, in order
to identify the possible obstructions to hyperbolicity, and we prove that the cone-off of
a CAT(0) cube complex, killing these obstructions, turns out to be hyperbolic. More
precisely, we first prove:
Theorem 1.3. Let X be a CAT(0) cube complex. The following are equivalent:
(i) X is hyperbolic;
(ii) the flat rectangles in X are uniformly thin;
(iii) X is finite-dimensional and the grid of hyperplanes in X are uniformly thin.
A flat rectangle is a combinatorially geodesic subcomplex isomorphic to some square
complex [0, a] × [0, b]; it is L-thin if min(a, b) ≤ L and L-thick if a, b ≥ L. A grid
of hyperplanes is the data of two families of hyperplanes V = {V1 , . . . , Vp } and H =
{H1 , . . . , Hq }, such that any Vi is transverse to any Hj , and any Vi (resp. Hj ) separates
Vi−1 and Vi+1 (resp. Hj−1 and Hj+1 ); such a grid is said δ-thin if min(#V, #H) ≤ δ.
Notice that an (n, m)-grid of hyperplanes is precisely the crossing graph of [0, n]×[0, m] ⊂
R2 endowed with its canonical structure of square complex.
Then, we notice that a cone-off “killing” the flat rectangles is hyperbolic:
Theorem 1.4. Let X be a CAT(0) cube complex and Y a cone-off of X over a collection
of combinatorially convex subcomplexes. If the thick flat rectangles of X are uniformly
bounded in Y , then Y is hyperbolic.
Our first application concerns weak relative hyperbolicity. In [Hag14], Hagen associates to any CAT(0) cube complex a hyperbolic graph, namely its contact graph, and
in particular he deduces that cubulable groups are weakly hyperbolic relatively to the
hyperplane stabilizers. Up to a quasi-isometry, looking at the contact graph amounts
2

to cone-off the (neighborhood of the) hyperplanes of the cube complex. Using Theorem
1.4, we show that it is possible to choose the hyperplanes we have to cone-off. More
precisely, for every n ≥ 0, we define a class of hyperplanes, the n-combinatorially contracting hyperplanes, and we prove that the cone-off Γn X of X over the hyperplanes
which are not n-combinatorially contracting is hyperbolic. In particular, we are able to
slightly improve the result of Hagen:
Corollary 1.5. Let G be a group acting geometrically on a CAT(0) cube complex X.
Then G is weakly hyperbolic relatively to the stabilizers of the non-contracting hyperplanes of X.
For instance, applied to the class of right-angled Coxeter groups, we obtain:
Proposition 1.6. Let Γ be a finite graph. The right-angled Coxeter group C(Γ) is
weakly hyperbolic relatively to each of the following collection of subgroups
• {C(Γ1 ∗ Γ2 ) | Γ1 ∗ Γ2 ⊂ Γ with Γ1 , Γ2 not complete},
• {hstar(u)i | u ∈ (Γ)},
where (Γ) denotes the set of vertices of Γ which belong to an induced square.
Afterwards, we focus on strong relative hyperbolicity. Therefore, we first prove
a criterion to determine when the usual cone-off of a CAT(0) cube complex is fine.
Explicitly,
Theorem 1.7. Let X be a uniformly locally finite CAT(0) cube complex and Y a usual
cone-off of X over a collection Q of combinatorially convex subcomplexes. If Q is locally
finite (ie., there exist only finitely-many subcomplexes of Q containing a given edge of X)
and if there exists a constant C ≥ 0 such that two subcomplexes of Q are both intersected
by at most C hyperplanes, then Y is fine. Conversely, if Q is not locally finite or if it
contains two subcomplexes both intersected by infinitely many hyperplanes, then Y is not
fine.
Thus, combining this criterion with Theorem 1.4, we are able to reprove the characterization of the strong relative hyperbolicity of right-angled Coxeter groups stated in
[BHSC13, Theorem I]. Given a finite graph Γ, we find a collection of subgraphs J∞ (Γ)
such that:
Theorem 1.8. The right-angled Coxeter group C(Γ) is relatively hyperbolic if and only
if J∞ (Γ) 6= {Γ}. If so, then C(Γ) is hyperbolic relatively to {C(Λ) | Λ ∈ J∞ (Γ)}.
According to Theorem 1.3, infinite-dimensional CAT(0) cube complexes cannot be
hyperbolic. Thus, a natural cone-off to look at is the cone-off over the high dimensional cubes. In fact, up to a quasi-isometry, it amounts to consider the distance d∞
corresponding to the `∞ -norm on each cube. Thus, considering the distance d∞ allows
us to introduce hyperbolicity in infinite dimensions. In fact, since loosely speaking the
cone-off kills the dimension, the criterion suggested by Theorem 1.3 is precisely what
we get:
Theorem 1.9. Let X be a CAT(0) cube complex. Then (X, d∞ ) is hyperbolic if and
only if the grid of hyperplanes in X are uniformly thin.
For instance, infinite-dimensional CAT(0) cube complexes in which the grid of hyperplanes are uniformly thin naturally appear in some infinitely-presented groups which
are “limits” of hyperbolic groups; for example, they include the cubulations of infinitelypresented C 0 (1/6) or C 0 (1/4) − T (4) groups [Wis04]. In order to exhibit a negativelycurved behaviour of such groups, we determine when the action on this hyperbolic space
satisfies an acylindrical condition.
3

Theorem 1.10. Let G be a group acting on a complete CAT(0) cube complex X. Suppose (X, d∞ ) hyperbolic. The following statements are equivalent:
(i) for every d ≥ 0, there exists R ≥ 0 such that, for every vertices x, y ∈ X,
d∞ (x, y) ≥ R ⇒ #{g ∈ G | d∞ (x, gx), d∞ (y, gy) ≤ d} < +∞;
(ii) there exists R ≥ 0 such that, for every vertices x, y ∈ X,
d∞ (x, y) ≥ R ⇒ #{g ∈ G | gx = x, gy = y} < +∞;
(iii) there exists R ≥ 0 such that, for any hyperplanes J1 , J2 separated by at least R
pairwise disjoint hyperplanes, stab(J1 ) ∩ stab(J2 ) is finite.
Such an action is useful to prove the acylindrical hyperbolicity of the group (see
Section 8 for precise definitions), since any loxodromic isometry of G turns out to be
WPD. For instance, we are able to deduce:
Theorem 1.11. Infinitely-presented C 0 (1/4)−T (4) groups are acylindrically hyperbolic.
Theorem 1.12. Let G = G1 ∗ · · · ∗ Gn be a finitely-generated free product and R ⊂ G a
family satisfying the condition C 0 (1/4) − T (4). Then the quotient Q = G/hhRii is either
virtually cyclic or acylindrically hyperbolic.
The first result may be compared with [GS14], where it is proved, in particular, that
infinitely-presented C(7) groups are acylindrically hyperbolic.
This paper is organised as follows. Section 2 contains the necessary prelimiaries on
CAT(0) cube complexes, including combinatorial projections and disc diagrams. Sections 3 and 4 are respectively dedicated to the proofs of Theorems 1.3 and 1.4. In
Section 5, we prove the applications toward weak and strong relative hyperbolicities
that we mentionned above. Finally, Section 6 contains the proof of 1.9, Section 7 the
proof of Theorem 1.10, and these two results are combined in Section 8 to deduce Theorem 1.11 and Theorem 1.12.

2

Preliminaries

A cube complex is a CW complex constructed by gluing together cubes of arbitrary
(finite) dimension by isometries along their faces. Furthermore, it is nonpositively curved
if the link of any of its vertices is a simplicial flag complex (ie., n + 1 vertices span a
n-simplex if and only if they are pairwise adjacent), and CAT(0) if it is nonpositively
curved and simply-connected. See [BH99, page 111] for more information.
Alternatively, CAT(0) cube complexes may be described by their 1-skeletons. Indeed,
Chepoi notices in [Che00] that the class of graphs appearing as 1-skeletons of CAT(0)
cube complexes coincides with the class of median graphs, which we now define.
Let Γ be a graph. If x, y, z ∈ Γ are three vertices, a vertex m is called a median point
of x, y, z whenever
d(x, y) = d(x, m) + d(m, y), d(x, z) = d(x, m) + d(m, z), d(y, z) = d(y, m) + d(m, z).
Notice that, for every geodesics [x, m], [y, m] and [z, m], the concatenations [x, m]∪[m, y],
[x, m] ∪ [m, z] and [y, m] ∪ [m, z] are also geodesics; furthermore, if [x, y], [y, z] and [x, z]
are geodesics, then any vertex of [x, y] ∩ [y, z] ∩ [x, z] is a median point of x, y, z.
The graph Γ is median if every triple (x, y, z) of pairwise distinct vertices admits a
unique median point, denoted by m(x, y, z).
4

Theorem 2.1. [Che00, Theorem 6.1] A graph is median if and only if it is the 1-skeleton
of a CAT(0) cube complex.
A fundamental feature of cube complexes is the notion of hyperplane. Let X be a
nonpositively curved cube complex. Formally, a hyperplane J is an equivalence class of
edges, where two edges e and f are equivalent whenever there exists a sequence of edges
e = e0 , e1 , . . . , en−1 , en = f where ei and ei+1 are parallel sides of some square in X.
Notice that a hyperplane is uniquely determined by one of its edges, so if e ∈ J we say
that J is the hyperplane dual to e. Geometrically, a hyperplane J is rather thought of
as the union of the midcubes transverse to the edges belonging to J.

The neighborhood N (J) of a hyperplane J is the smallest subcomplex of X containing
J, i.e., the union of the cubes intersecting J. In the following, ∂N (J) will denote
the union of the cubes of X contained in N (J) but not intersecting J, and X\\J =
(X\N (J)) ∪ ∂N (J). Notice that N (J) and X\\J are subcomplexes of X.
Theorem 2.2. [Sag95, Theorem 4.10] Let X be a CAT(0) cube complex and J a hyperplane. Then X\\J has exactly two connected components.
The two connected components of X\\J will be refered to as the halfspaces associated
to the hyperplane J.
Distances `p . There exist several natural metrics on a CAT(0) cube complex. For
example, for any p ∈ (0, +∞), the `p -norm defined on each cube can be extended to a
distance defined on the whole complex, the `p -metric. Usually, the `1 -metric is refered
to as the combinatorial distance and the `2 -metric as the CAT(0) distance. Indeed, a
CAT(0) cube complex endowed with its CAT(0) distance turns out to be a CAT(0) space
[Lea13, Theorem C.9], and the combinatorial distance between two vertices corresponds
to the graph metric associated to the 1-skeleton X (1) . In particular, combinatorial
geodesics are edge-paths of minimal length, and a subcomplex is combinatorially convex
if it contains any combinatorial geodesic between two of its points.
In fact, the combinatorial metric and the hyperplanes are strongly linked together:
the combinatorial distance between two vertices corresponds exactly to the number of
hyperplanes separating them [Hag08, Theorem 2.7], and
Theorem 2.3. [Hag08, Corollary 2.16] Let X be a CAT(0) cube complex and J a
hyperplane. The two components of X\\J are combinatorially convex, as well as the
components of ∂N (J).
The `∞ -metric, denoted by d∞ , is also of particular interest. Alternatively, given
a CAT(0) cube complex X, the distance d∞ between two vertices corresponds to the
distance associated to the graph obtained from X (1) by adding an edge between two
vertices whenever they belong to a common cube. Nevertheless, the distance we obtain
stays strongly related to the combinatorial structure of X:
5

Proposition 2.4. [BvdV91, Corollary 2.5] Let X be a CAT(0) cube complex and x, y ∈
X two vertices. Then d∞ (x, y) is the maximal number of pairwise disjoint hyperplanes
separating x and y.
Proposition 2.5. In a CAT(0) cube complex, any half-space is geodesic with respect to
the distance d∞ .
Proof. Let X be a CAT(0) cube complex, D a half-space and x, y ∈ D two vertices.
To conclude, it is sufficient to prove that a d∞ -geodesic between x and y intersecting
a minimal number of half-spaces must be included into D. Let γ be such a geodesic
and suppose by contradiction that γ is not included into D. In particular, there exists
a hyperplane J intersected twice by γ and a subsegment γ0 ⊂ γ with endpoints in
N (J) such that γ0 does not intersect twice any hyperplane. By linking two consecutive
vertices of γ0 by a combinatorial geodesic, we produce a combatorial path γ0 with the
same endpoints as γ0 ; moreover, since γ0 does not intersect twice any hyperplane, so
does γ0 , ie., γ0 is a combinatorial geodesic. This implies that
γ0 ⊂ γ0 ⊂ ∂N (J),
by the combinatorial convexity of the components of ∂N (J). Let γ 0 be the d∞ -path
obtained from γ by replacing the subsegment γ0 with its image in N (J) by the reflection
with respect to J. Noticing that length(γ 0 ) ≤ length(γ), we deduce that γ 0 is a new
d∞ -geodesic with the same endpoints as γ. On the other hand, the number of halfspaces intersected by γ 0 is strictly smaller than the same number for γ, a contradiction.
Therefore, γ is included into D. 
Combinatorial projection In CAT(0) spaces, and so in particular in CAT(0) cube
complexes with respect to the CAT(0) distance, the existence of a well-defined projection
onto a given convex subspace provides a useful tool. Similarly, with respect to the
combinatorial distance, it is possible to introduce a combinatorial projection onto a
combinatorially convex subcomplex, defined by the following result.
Proposition 2.6. [Gen15, Lemma 1.2.3] Let X be a CAT(0) cube complex, C ⊂ X be
a combinatorially convex subspace and x ∈ X\C be a vertex. Then there exists a unique
vertex y ∈ C minimizing the distance to x. Moreover, for any vertex of C, there exists
a combinatorial geodesic from it to x passing through y.
For example, we are able to deduce the following result:
Proposition 2.7. Let X be a CAT(0) cube complex and C1 , C2 ⊂ X two combinatorially
convex subspaces. If x ∈ C1 and y ∈ C2 minimize the distance between C1 and C2 then
the hyperplanes separating x and y are precisely those separating C1 and C2 .
Lemma 2.8. Let X be a CAT(0) cube complex and N ⊂ X a combinatorially convex
subspace. Let p : X → N denote the combinatorial projection onto N . Then every
hyperplane separating x and p(x) separates x and N .
Proof. According to Proposition 2.6, for all z ∈ N there exists a combinatorial geodesic
γ between x and z passing through p(x). Of course, any hyperplane separating x and
p(x) meets γ, so any such hyperplane cannot meet [p(x), z] ⊂ γ. Thus, we have proved
that no hyperplane separating x and p(x) separates p(x) and some vertex of N . 
Proof of Proposition 2.7. Clearly, a hyperplane separating C1 and C2 separates x
and y. Conversely, let J be a hyperplane separating x and y, and let p : X → C1 and
q : X → C2 denote the combinatorial projection onto C1 and C2 respectively. Because x
and y minimize the distance between C1 and C2 , it follows that x = p(y) and y = q(x).
6

According to Lemma 2.8, we deduce that J separates x and C2 , and y and C1 . Thus, J
separates C1 and C2 . 
We conclude this paragraph with this last result:
Proposition 2.9. Let X be a CAT(0) cube complex and C1 , C2 ⊂ X two combinatorially
convex subcomplexes. Let p : X → C2 denote the combinatorial projection onto C2 . Then
p(C1 ) is a geodesic subcomplex of C2 . Moreover, the hyperplanes intersecting p(C1 ) are
precisely those which intersect both C1 and C2 .
Lemma 2.10. Let X be a CAT(0) cube complex and C a combinatorially convex subcomplex. Let p : X → C denote the combinatorial projection onto C. For any vertices
x, y ∈ X and any hyperplane J separating p(x) and p(y), J separates x and y. In
particular, d(p(x), p(y)) ≤ d(x, y).
Proof. Because the hyperplane J separates two vertices of C, namely p(x) and p(y),
it follows from Lemma 2.8 that J cannot separate neither x and p(x) nor y and p(y).
Necessarily, J separates x and y. 
Proof of Proposition 2.9. Because d(p(x), p(y)) ≤ d(x, y), two adjacent vertices
in C1 are sent by p onto the same vertex or onto two adjacent vertices. Thus, the
image by p of a combinatorial geodesic γ in C1 between two vertices x and y defines a
combinatorial path p(γ) in C2 between p(x) and p(y). Using the previous lemma, we
deduce that the length of p(γ) is precisely the number of hyperplanes separating p(x)
and p(y). Therefore, p(γ) is a combinatorial geodesic between p(x) and p(y) included
into p(C1 ). 
Disc diagrams A fundamental tool to study CAT(0) cube complexes is the theory
of disc diagrams. For example, they were extensively used by Sageev in [Sag95] and by
Wise in [Wis12]. The rest of this section is dedicated to basic definitions and properties
of disc diagrams.
Definition 2.11. Let X be a nonpostively curved cube complex. A disc diagram is a
continuous combinatorial map D → X, where D is a finite contractible square complex
with a fixed topological embedding into S2 ; notice that D may be non-degenerated, ie.,
homeomorphic to a disc, or may be degenerated. In particular, the complement of D in
S2 is a 2-cell, whose attaching map will be refered to as the boundary path ∂D → X
of D → X; it is a combinatorial path. The area of D → X, denoted by Area(D),
corresponds to the number of squares of D.
Given a combinatorial closed path P → X, we say that a disc diagram D → X is
bounded by P → X if there exists an isomorphism P → ∂D such that the following
diagram is commutative:
∂D
7/ X
O
P
According to a classical argument due to van Kampen [VK33], there exists a disc diagram
bounded by a given combinatorial closed path if and only if this path is null-homotopic.
Thus, if X is a CAT(0) cube complex, then any combinatorial closed path bounds a disc
diagram.
As a square complex, a disc diagram contains hyperplanes: they are called dual
curves. Equivalently, they correspond to the connected components of the reciprocal
images of the hyperplanes of X. Given a disc diagram D → X, a nogon is a dual curve
7

homeomorphic to a circle; a monogon is a subpath, of a self-intersecting dual curve,
homeomorphic to a circle; an oscugon is a subpath of a dual curve whose endpoints are
the midpoints of two adjacent edges; a bigon is a pair of dual curves intersecting into
two different squares.

Figure 1: From left to right: a nogon, a monogon, an oscugon and a bigon.
Theorem 2.12. [Wis12, Lemma 2.2] Let X be a nonpositively curved cube complex and
D → X a disc diagram. If D contains a nogon, a monogon, a bigon or an oscugon,
then there exists a new disc diagram D0 → X such that:
(i) D0 is bounded by ∂D,
(ii) Area(D0 ) ≤ Aire(D) − 2.
Let X be a CAT(0) cube complex. A cycle of subcomplexes C is a sequence of
subcomplexes C1 , . . . , Cr such that C1 ∩Cr 6= ∅ and Ci ∩Ci+1 6= ∅ for every 1 ≤ i ≤ r −1.
A disc diagram D → X is bounded by C if ∂D → X can be written as the concatenation
of r combinatorial geodesics P1 , . . . , Pr → X such that Pi ⊂ Ci for every 1 ≤ i ≤ r. The
complexity of such a disc diagram is defined by the couple (Area(D), length(∂D)), and
a disc diagram bounded by C will be of minimal complexity if its complexity is minimal
with respect to the lexicographic order among all the possible disc diagrams bounded
by C (allowing modifications of the paths Pi ).
Our main technical result on disc diagrams is the following, inspired by a result due
to Hagen [Hag14, Lemma 2.11]. The arguments used here are essentially the same, and
mainly come from [Wis12].
Theorem 2.13. Let X be a CAT(0) cube complex, C = (C1 , . . . , Cr ) a cycle of subcomplexes, and D → X a disc diagram bounded by C. For convenience, write ∂D as
the concatenation of r combinatorial geodesics P1 , . . . , Pr → X with Pi ⊂ Ci for every
1 ≤ i ≤ r. If the complexity of D → X is minimal, then:
(i) if Ci is combinatorially convex, two dual curves intersecting Pi are disjoint;
(ii) if Ci and Ci+1 are combinatorially convex, no dual curve intersects both Pi and
Pi+1 .
Proof. Suppose that Ci is combinatorially convex and that there exist two transverse
dual curves c1 and c2 intersecting Pi . Any dual curve intersecting Pi between c1 and
c2 necessarily intersects either c1 or c2 , so we may suppose without loss of generality
that c1 and c2 intersect Pi along two adjacent edges. If these two edges bound a square
inside D, then we replace Pi with Pi0 as follows:

8

Because Ci is combinatorially convex, Pi0 ⊂ Ci . Thus, we have constructed a new
disc diagram D0 → X bounded by C, such that Area(D0 ) = Area(D) − 1, so that
c(D0 ) < c(D). This contradicts the minimality of the complexity of D. Now, suppose
that the edges of Pi dual to c1 and c2 do not bound a square inside D. Because the
hyperplanes of X do not inter-osculate, the images of these two edges bound a square
inside X, so that it is possible to construct a new disc diagram D0 → X as follows:

Similarly, by the combinatorial convexity of Ci , we have Pi0 ⊂ Ci so that D0 → X is
bounded by C. Noticing that D0 contains a bigon, Theorem 2.12 allows us to construct
a new disc diagram D00 → X bounded by ∂D0 and satisfying Area(D00 ) ≤ Area(D0 ) − 2.
Therefore, we have Aire(D00 ) ≤ Aire(D) − 1, so that c(D00 ) < c(D) which contradicts
the minimality of the complexity of D. This concludes the proof of the point (i).
In fact, we have proved a more precise statement which will be useful later:
Fact 2.14. Let X be a CAT(0) cube complex, C = (C1 , . . . , Cr ) a cycle of subcomplexes,
and D → X a disc diagram bounded by C. For convenience, write ∂D as the concatenation of r combinatorial geodesics P1 , . . . , Pr → X with Pi ⊂ Ci for every 1 ≤ i ≤ r. If
there exists an index 1 ≤ i ≤ r such that there exist two transverse dual curves intersecting Ci , then there exists a new disc diagram D0 → X satisfying Area(D0 ) ≤ Area(D) − 2
and whose path boundary is the concatenation of P1 , . . . , Pi−1 , Pi0 , Pi+1 , . . . , Pr where Pi0
is a combinatorial geodesic with the same endpoints as Pi .
Now, suppose that Ci and Ci+1 are combinatorially convex and that there exists a
dual curve c intersecting both Pi and Pi+1 ; let ei and ei+1 denote respectively the
edges of Pi and Pi+1 dual to c. Because two dual curves intersecting either Pi or Pi+1
are necessarily disjoint according to the point (i), no dual curve intersecting Pi ∪ Pi+1
between ei and ei+1 can intersect c, so we may suppose without loss of generality that ei
and ei+1 are adjacent. If ei 6= ei+1 , then D contains an oscugon so that it is possible to
decrease the area of D without disturbing its boundary path according to Theorem 2.12,
contradicting the minimality of the complexity of D. Thus, ei = ei+1 . Now, replace Pi
0
and Pi+1 respectively with Pi0 and Pi+1
as follows:

9

We have constructed a new disc diagram D0 → X bounded by C and satisfying Area(D0 ) =
Area(D), length(∂D0 ) = length(∂D) − 1. In particular, we have c(D0 ) < c(D), contradicting the minimality of the complexity of D. 
We conclude the section by a last result on disc diagrams which will be useful later.
Proposition 2.15. Let X be a CAT(0) cube complex and D → X a disc diagram which
does not contain any bigon. With respect to the combinatorial metrics, ϕ : D → X is
an isometric embedding if and only if every hyperplane of X induces at most one dual
curve of D.
Lemma 2.16. Let D → X be a disc diagram which does not contain any bigon and
x, y ∈ D two vertices. The combinatorial distance between x and y in D is equal to the
number of dual curves separating x and y.
Proof. Let c(x, y) denote the number of dual curves separating x and y. Of course,
any combinatorial path between x and y must intersect each of the c(x, y) dual curves
separating x and y, hence d(x, y) ≥ c(x, y). Conversely, to prove that d(x, y) ≤ c(x, y),
it is sufficient to show that any combinatorial path of minimal length between x and y
intersects each dual curve at most once. Let us consider a combinatorial path γ between
x and y which intersects a dual curve at least twice; in particular, there exist two edges
e and f of this path dual to the same dual curve. We choose this dual curve innermost,
so that, if γ2 denotes the subpath of γ between e and f , then no dual curve intersects
γ2 twice. Let γ0 , γ1 denote the two combinatorial paths fellow-traveling our dual curve
between the endpoints of e and f .

In particular, any dual curve intersecting γ2 intersects γ1 . Conversely, because D does
not contain any bigon, any dual curve intersecting γ1 intersects necessarily γ2 . Consequently, length(γ1 ) = length(γ2 ). By replacing the subpath e ∪ γ2 ∪ f with γ0 in our
path, we get a new combinatorial path whose length is smaller, since
length(e ∪ γ2 ∪ f ) = length(γ2 ) + 2 = length(γ1 ) + 2 = length(γ0 ) + 2 > length(γ0 ).
We have proved that any combinatorial path between x and y which intersects a dual
curve twice may be shortened. This concludes the proof. 
10

Proof of Proposition 2.15. Suppose that there exist two dual curves c1 and c2
induced by the same hyperplane of X. Let x, y ∈ D be two vertices separated by c1
and c2 . The combinatorial distance between ϕ(x) and ϕ(y) corresponds to the number
of hyperplanes separating them; each of these hyperplanes induces a dual curve in D
separating x and y, so there exist at least d(ϕ(x), ϕ(y)) dual curves separating x and y,
hence d(x, y) ≥ d(ϕ(x), ϕ(y)). On the other hand, we know that at least two of these
dual curves, namely c1 and c2 , are induced by the same hyperplane, hence d(x, y) >
d(ϕ(x), ϕ(y)). Consequently, ϕ : D → X is not an isometric embedding.
Conversely, suppose that each hyperplane of X induces at most one dual curve on D.
Now fix two vertices x, y ∈ D together with a combinatorial path γ of minimal length
between x and y in D. According to the previous lemma, any dual curve intersects γ at
most once. Thus, the only possibility for a hyperplane of X to intersect ϕ(γ) twice is
to induce two different dual curve intersecting γ, which is impossible by our hypothesis.
We conclude that ϕ(γ) is a combinatorial geodesic in X, ie., d(ϕ(x), ϕ(y)) ≥ d(x, y). We
have already noticed in the previous paragraph the inequality d(x, y) ≥ d(ϕ(x), ϕ(y)),
so d(x, y) = d(ϕ(x), ϕ(y)). Consequently, ϕ : D → X is an isometric embedding. 
Corollary 2.17. Let X be a CAT(0) cube complex and C a cycle of four combinatorially
convex subcomplexes. If D → X is a disc diagram of minimal complexity bounded by C,
then D is combinatorially isometric to a rectangle [0, a] × [0, b] ⊂ R2 and D → X is an
isometric embedding.
Proof. By definition, the boundary path ∂D → X can be written as the concatenation
of four combinatorial geodesics P1 , P2 , P3 , P4 → X. It follows directly from Theorem
2.13 that two dual curves intersecting Pi are necessarily disjoint and that any dual
curve intersecting Pi necessarily intersects Pi+2 (taking i modulo 4). In particular,
length(P1 ) = length(P3 ) and length(P2 ) = length(P4 ); let us say that a dual curve
intersecting P1 (resp. P2 ) is vertical (resp. horizontal) and let x0 denote the common
endpoint of P1 and P4 . Now, the map x 7→ (v(x), h(x)), where v(x) (resp. h(x)) denotes
the number of vertical (resp. horizontal) dual curves separating x0 and x, defines a
combinatorial isometric embedding D ,→ Z2 and its image is a rectangle. Finally, it
follows easily from Proposition 2.15 that D → X is an isometric embedding: two dual
curves in D either intersects the same Pi , so they are necessarily induced by two different
hyperplanes since Pi is a combinatorial geodesic, or they are transverse, so that they
map onto two transverse hyperplanes, which are necessarily distinct since a CAT(0)
cube complex does not contain self-intersecting hyperplanes. 

3

Hyperbolicity with respect to the `1 and `2 metrics

In this section, we are interested in determining when a CAT(0) cube complex is hyperbolic with respect to the combinatorial or CAT(0) distance. It is worth noticing
that an n-cube contains a geodesic triangle which is not (n − 1)-thin with respect to
√
the combinatorial distance, and a geodesic triangle which is not ( n/2 − 1)-thin with
respect to the CAT(0) metric; therefore, an infinite-dimensional CAT(0) cube complex
is hyperbolic neither with respect to the combinatorial distance nor with respect to
the CAT(0) distance. Thus, the hyperbolicity happens only in finite dimension, where
these two distances are quasi-isometric, so that being hyperbolic does not depend on
the metric we choose. For convenience, we will use the combinatorial distance.
Definition 3.1. A grid of hyperplanes is the data of two families of hyperplanes V =
{V1 , . . . , Vp } and H = {H1 , . . . , Hq }, such that any Vi is transverse to any Hj , and any
Vi (resp. Hj ) separates Vi−1 and Vi+1 (resp. Hj−1 and Hj+1 ); such a grid is said δ-thin
if min(#V, #H) ≤ δ.
11

Definition 3.2. Let X be a CAT(0) cube complex. A flat rectangle is a combinatorially
geodesic subcomplex of X isomorphic to some rectangle [0, a] × [0, b] ⊂ R2 , with a, b ≥ 1,
endowed with its canonical structure of square complex; it is L-thick if a, b ≥ L and
L-thin if min(a, b) ≤ L.
Our main criterion is the following:
Theorem 3.3. Let X be a CAT(0) cube complex. The following are equivalent:
(i) X is hyperbolic;
(ii) the flat rectangles in X are uniformly thin;
(iii) X is finite-dimensional and the grids of hyperplanes of X are uniformly thin.
Remark 3.4. The equivalence (i) ⇔ (iii) may be compared with [Hag14, Theorem
7.3], where Hagen proves that a cocompact CAT(0) cube complex is hyperbolic if and
only if its crossing graph does not contain a complete bipartite subgraph K∞,∞ . Recall
that the crossing graph of a CAT(0) cube complex X is the graph whose vertices are
the hyperplanes of X and whose edges link two transverse hyperplanes.
In fact, a simple criterion of hyperbolicity for median graphs has already been proved by
Sigarreta, expressed in terms of bigons [Sig13]. (Notice that Papasoglu proved that the
same statement holds for every graph [Pap95], but the argument of Sigarreta, reproduced
below for completeness, is completely elementary.) A fortiori, this produces a criterion
of hyperbolicity for CAT(0) cube complexes, which will be used to prove the implication
(iii) ⇒ (i) of Theorem 3.3.
Definition 3.5. A bigon is a pair of combinatorial geodesics γ1 , γ2 with the same endpoints. It is δ-thin if the Hausdorff distance between γ1 and γ2 is at most δ.
Lemma 3.6. [Sig13, Theorem 2.5] A CAT(0) cube complex is hyperbolic if and only if
its bigons are uniformly thin.
Proof. It is known that, in a δ-hyperbolic geodesic space, the Hausdorff distance
between two geodesics with the same endpoints is bounded by a constant C(δ) depending
only on δ. Therefore, the bigons in a hyperbolic graph are uniformly thin.
Conversely, suppose there exists some δ > 0 such that the bigons of X are all δ-thin.
Let [x, y, z] be a geodesic triangle. Let m denote the median point of {x, y, z}. Let p ∈
[x, y, z] be a point on our triangle, say p ∈ [x, y]. Because the bigon {[x, y], [x, m]∪[m, y]}
is δ-thin, there exists a point p0 ∈ [x, m] ∪ [m, y], say p0 ∈ [x, m], such that d(p, p0 ) ≤ δ.
Again, because the bigon {[x, m] ∪ [m, z] ∪ [x, z]} is δ-thin, there exists a point p00 ∈ [x, z]
such that d(p0 , p00 ) ≤ δ. Thus,
d(p, [x, z]) ≤ d(p, p00 ) ≤ d(p, p0 ) + d(p0 , p00 ) ≤ 2δ.
Therefore, the triangle [x, y, z] is 2δ-thin. We conclude that X is 2δ-hyperbolic. 
Let Ram(d) denote the Ramsey number defined by: if K is a complete graph with at
least Ram(d) vertices, then any label of the vertices of K with two colors contains a
unicolor subgraph containing at least d+1 vertices. A simple and well-known application
of Ramsey numbers in the context of CAT(0) cube complexes is:
Lemma 3.7. Let X be a CAT(0) cube complex and let H be a collection of hyperplanes
which does not contain d+1 pairwise transverse hyperplanes. (For instance, this happens
when dim X ≤ d.) If H has at least Ram(d) hyperplanes, then its contains a family of
at least d + 1 pairwise disjoint hyperplanes.
12

Proof. Let H be a collection of at least Ram(d) hyperplanes. Let K be the complete
graph whose set of vertices is H and whose edges are red (resp. blue) if they link two
transverse (resp. disjoint) hyperplanes. By definition of Ram(d), K contains a unicolor
subgraph with at least d + 1 vertices, ie., H has to contain at least d + 1 hyperplanes
which are either pairwise transverse or pairwise disjoint. But the first possibility is
excluded by our hypotheses, so that our lemma follows. 
Proof of Theorem 3.3. The implication (i) ⇒ (ii) is clear.
Let us prove (ii) ⇒ (iii). Suppose that all flat rectangles in X are L-thin and let
(V = {V1 , . . . , Vn }, H = {H1 , . . . , Hm }) be an (n, m)-grid of hyperplanes. In particular,
C = (N (V1 ), N (H1 ), N (Vn ), N (Hm )) defines a cycle of subcomplexes. Let D → X be
a disc diagram of minimal complexity bounded by C. According to Corollary 2.17, we
find a combinatorial isometric embedding [0, a] × [0, b] ,→ X. Furthermore, because
V2 , . . . , Vn−1 (resp. H1 , . . . , Hm−1 ) separate V1 and Vn (resp. H1 and Hm ), we deduce
that a ≥ n − 1 (resp. b ≥ m − 1). We conclude that
min(n, m) ≤ 1 + min(a, b) ≤ L + 1.
Thus, the grid of hyperplanes in X are uniformly thin. Finally, to prove that X is
necessarily finite-dimensional, it is sufficient to notice that the cube [0, 1]n contains a
flat rectangle which is not bn/2c-thin. For instance, if
ep (i) = ( 1, . . . , 1 ,
| {z }

0, . . . , 0

),

| {z }

i coordinates p−i coordinates

then the vertices (ebn/2c (i), en−bn/2c (j)) ∈ [0, 1]n , where 0 ≤ i ≤ bn/2c and 0 ≤ j ≤
n − bn/2c, span a flat rectangle [0, bn/2c] × [0, n − bn/2c] ,→ [0, 1]n .
Finally, we prove (iii) ⇒ (i). Suppose that dim X < +∞ and that the grid of hyperplanes in X are all C-thin for some C ≥ 1; without loss of generality, we may suppose
that C ≥ dim X. According to Lemma 3.6, it is sufficient to prove that the bigons
of X are uniformly thin in order to deduce that X is hyperbolic. So let γ1 , γ2 be two
combinatorial geodesics with the same endpoints x, y. If a ∈ γ1 , let b ∈ γ2 denote the
vertex of γ2 satisfying d(x, a) = d(x, b). Let H1 denote the set of hyperplanes meeting
[x, a] ⊂ γ1 and separating a and b, H2 the set of hyperplanes meeting [x, b] ⊂ γ2 and
separating a and b, and finally H3 the set of hyperplanes separating x and {a, b}.

Because the hyperplanes of H1 separate b and {a, y}, and the hyperplanes of H2 separate
a and {b, y}, we deduce that any hyperplane of H1 is transverse to any hyperplane of
H2 . Now, notice that
#H3 + #H1 = d(x, a) = d(x, b) = #H3 + #H2 ,
whence #H1 = #H2 . Let p denote this common cardinality. If p ≥ Ram(C), then
Lemma 3.7 implies that H1 and H2 each contain a subfamily of at least C + 1 pairwise
disjoint hyperplanes, producing a (C + 1, C + 1)-grid of hyperplanes and thus contradicting our hypotheses. Therefore, p ≤ Ram(C). We conclude that
13

d(a, b) = #H1 + #H2 = 2p ≤ 2Ram(C).
We have proved that the bigons of X are all 2Ram(C)-thin, so that X is hyperbolic (in
fact, 4Ram(C)-hyperbolic). 
In particular, an immediate corollary of Theorem 3.3 is that a CAT(0) cube complex
whose transversality graph is uniformly locally finite is hyperbolic. In fact, we can even
prove more:
Proposition 3.8. A CAT(0) cube complex whose transversality graph is uniformly locally finite is quasi-isometric to a tree.
Proof. Let X be a CAT(0) cube complex whose transversality graph is uniformly locally
finite, ie., there exists a constant N such that any hyperplane of X is transverse to at
most N other hyperplanes. This implies that the diameter of any hyperplane J is at
most N . Indeed, if x, y are two vertices which belongs to the same connected component
of ∂N (J), then any hyperplane separating x and y is transverse to J, hence d(x, y) ≤ N .
Now, it is easy to apply the following criterion:
Bottleneck criterion. [Man05] A geodesic metric space (S, d) is quasi-isometric to a
tree if and only if there exists a constant δ > 0 such that, any pair of points x, y ∈ S
admits a midpoint m = m(x, y) with the property that any continuous path between x
and y intersects the ball B(m, δ).
Indeed, if x, y ∈ X are two points then fix any midpoint m = m(x, y) and let J denote a
hyperplane separating x and y such that m ∈ N (J). Now, let γ be any continous path
between x and y. Of course, γ has to pass through J, so that it finally intersects the
ball B(m, N ) ⊃ J. 
Remark 3.9. When X is a CAT(0) cube complex obtained by cubulating a codimensionone subgroup, Niblo characterized the (uniform) local finiteness of the transversality
graph of X from an algebraic viewpoint thanks to its splitting obstruction [Nib02, Proposition 7], and used it to establish a splitting result in the spirit of Stallings Theorem
[Nib02, Theorem B].
Remark 3.10. The hypothesis of uniform local finiteness in the statement of Proposition 3.8 cannot be weakened into a hypothesis of local finitess. Indeed, let X be a
subcomplex of R2 delimited by two combinatorial rays r = (rx , ry ) and ρ = (ρx , ρy ) with
• rx , ry , ρx , ρy : N → N,
• rx (0) = ry (0) = ρx (0) = ρy (0) = 0,
• rx (t), ry (t), ρx (t), ρy (t) −→ +∞,
t→+∞

• the Hausdorff distance between r and ρ is infinite.
Then the transversality graph of X is locally finite but X is not hyperbolic.

4

Killing flat rectangles

In this section, we prove that a cone-off of a CAT(0) cube complex “killing” the flat
rectangles of Theorem 3.3 is hyperbolic. This is the main result of this section.
Theorem 4.1. Let X be a CAT(0) cube complex and Y a cone-off of X over a collection
of combinatorially convex subcomplexes. If there exist some constants C, L ≥ 1 such that
the diameter of any L-thick flat rectangle of X is at most C in Y , then Y is δ-hyperbolic
for some δ depending only on C and L.
14

To prove the hyperbolicity, we will use the following criterion:
Lemma 4.2. Let X be a CAT(0) cube complex and Y a cone-off of X over a collection
of combinatorially convex subcomplexes. If there exists some C ≥ 1 such that the bigons
of X are all C-thin in Y , then Y is δ-hyperbolic for some δ depending only on C.
Proof. For any vertices x, y ∈ Y , let η(x, y) ⊂ Y denote the union of all the combinatorial geodesics between x and y in X. Now, we want to apply the following criterion,
due to Bowditch [Bow14, Proposition 3.1]:
Proposition 4.3. Let T be a graph and D ≥ 0. Suppose that a connected subgraph
η(x, y), containing x and y, is associated to any pair of vertices (x, y) ∈ T 2 such that:
• for any vertices x, y ∈ T , d(x, y) ≤ 1 implies diam η(x, y) ≤ D;
• for any vertices x, y, z ∈ T , we have η(x, y) ⊂ (η(x, z) ∪ η(z, y))+D .
Then T is δ-hyperbolic for some δ depending only on D.
The first condition is immediate: if dY (x, y) ≤ 1, either x = y and η(x, y) is a vertex;
or x and y are linked by an edge in X, so that they are the only vertices of η(x, y); or
x and y are linked by an edge in Y which does not belong to X, so that there exists a
combinatorially convex subcomplex Q ⊂ X containing x and y, and a fortiori η(x, y),
whose vertices are pairwise linked by an edge in Y . Therefore, dY (x, y) ≤ 1 implies
diamY η(x, y) ≤ 1.
Let x, y, z ∈ Y be three vertices. The second condition of Bowditch’s criterion is trivially
satisfied if x, y, z are not pairwise distinct. Thus, we may suppose without loss of
generality that they are pairwise distinct. Let m = m(x, y, z) denote their median vertex,
and fix three combinatorial geodesics [m, x], [m, y] and [m, z] in X. Let w ∈ η(x, y), ie., w
belongs to some combinatorial geodesic [x, y] between x and y in X. By our hypothesis,
the bigon {[x, m]∪[m, y], [x, y]} is C-thin in Y , so there exists a vertex w0 ∈ [x, m]∪[m, y]
such that dY (w, w0 ) ≤ C. Without loss of generality, say that w0 ∈ [x, m]; otherwise,
just switch the names of x and y. Similarly, the bigon {[x, m] ∪ [m, z], [x, z]} is C-thin
in Y , so there exists a vertex w00 ∈ [x, z] ⊂ η(x, z) such that dY (w0 , w00 ) ≤ C. Thus,
dY (w, η(x, z)) ≤ dY (w, w00 ) ≤ dY (w, w0 ) + dY (w0 , w00 ) ≤ 2C.
We have proved that η(x, y) ⊂ (η(x, z) ∪ η(z, y))+2C . This concludes the proof. 
Proof of Theorem 4.1. According to the previous lemma, it is sufficient to prove that
any bigon of X is max(2L, C)-thin in Y .
Let x, y ∈ Y be two vertices and γ1 , γ2 two combinatorial geodesics between x and y in X.
Fix some vertex z ∈ γ1 and let z 0 ∈ γ2 be the vertex of γ2 defined by dX (x, z 0 ) = dX (x, z).
We claim that dY (z, z 0 ) ≤ max(2L, C).

Let p (resp. q) denote the median vertex of {x, z, z 0 } (resp. {y, z, z 0 }), and let r (resp.
s) be the median vertex of {z, p, q} (resp. {z 0 , p, q}). Notice that
15

dX (x, z) = dX (x, p) + dX (p, z) = dX (x, p) + dX (p, r) + dX (r, z).
Therefore, if we fix three combinatorial geodesics [x, p], [p, r], [r, z], and if [x, z] denotes
the subsegment of γ1 between x and z, then {[x, p] ∪ [p, r] ∪ [r, z], [x, z]} is a bigon in
X. Thus, if J is a hyperplane separating z and r, necessarily J intersects [x, z] ⊂ γ1 .
The same argument with respect to y, r, q, z shows that J must intersect [z, y] ⊂ γ1 . In
particular, J intersects γ1 twice: this is impossible since γ1 is a combinatorial geodesic.
We conclude that no hyperplane separates z and r, ie., r = z. Similarly, we prove that
s = z 0 . In fact, we have proved:
Fact 4.4. For any bigon {γ1 , γ2 } with endpoints x, y and for any z ∈ γ1 , z 0 ∈ γ2 , we
have
m(z, m(x, z, z 0 ), m(y, z, z 0 )) = z.
Consequently, for any choices of four combinatorial geodesics [p, z], [z, q], [q, z 0 ] and
[z 0 , p], the concatenations [p, z] ∪ [z, q], [z, q] ∪ [q, z 0 ], [q, z 0 ] ∪ [z 0 , p] and [z 0 , p] ∪ [p, z] are
combinatorial geodesics.
Let D → X be a disc diagram bounded by a 4-gon P = P (p, z, q, z 0 ), ie., bounded by
four combinatorial geodesics [p, z], [z, q], [q, z 0 ] and [z 0 , p]; suppose that its complexity
is minimal among all the disc diagrams bounded by a similar 4-gon. According to
Fact 2.14, if there exist two transverse dual curves in D both intersecting the same
side of P , then it is possible to find a new disc diagram D0 → X bounded by another
4-gon P 0 = P 0 (p, z, q, z 0 ) such that Area(D0 ) < Area(D). Thus, two dual curves in
D intersecting the same side of P are necessarily disjoint. Furthermore, because the
concatenations [p, z]∪[z, q], [z, q]∪[q, z 0 ], [q, z 0 ]∪[z 0 , p] and [z 0 , p]∪[p, z] are combinatorial
geodesics, no dual curve in D intersects two adjacent sides of P . Consequently, D is a
rectangle. In particular, two dual curves in D either are transverse or intersect the same
side of P , so that a hyperplane of X cannot induce two dual curves in D: we deduce
from Proposition 2.15 that D → X is an isometric embedding.
Let ` = dX (p, z) = dX (q, z 0 ). Notice that
dX (p, z 0 ) = dX (x, z 0 ) − dX (x, p) = dX (x, z) − dX (x, p) = dX (p, z) = `,
and similarly dX (q, z) = `. Thus, D defines an `-thick flat rectangle in X. Two cases may
happen: either ` < L and dY (z, z 0 ) ≤ dX (z, z 0 ) = 2` < 2L; or ` ≥ L and dY (z, z 0 ) ≤ C.
Therefore, dY (z, z 0 ) ≤ max(2L, C). 

5
5.1

Relative hyperbolicity
Weak relative hyperbolicity

In the previous section, we have seen how a CAT(0) cube complex becomes hyperbolic
when some of its subcomplexes are coned-off. In this section, we apply this criterion
to cone-off hyperplanes and finally deduce a weak relative hyperbolicity of cubulable
groups.
Definition 5.1. Let X be a CAT(0) cube complex. A hyperplane J is n-combinatorially
contracting if dim J < n and J does not belong to an (n, n)-grid of hyperplanes in X.
Definition 5.2. Let X be a CAT(0) cube complex and n ≥ 0 an integer. The n-th
contracting graph of X, denobed by Γn X, is the cone-off of X over the neighborhoods
of the hyperplanes which are not n-combinatorially contracting.
If X has finitely-many orbits of hyperplanes under the action of Aut(X), then the
sequence (Γn X) is eventually constant to some graph Γ∞ X, which will be called the
contracting graph of X.
16

Proposition 5.3. Let X be a CAT(0) cube complex. Its n-th contracting graph Γn X is
δ(n)-hyperbolic, where δ(n) depends only on n.
Proof. According to Theorem 4.1, it is sufficient to prove that any Ram(n)-thick flat
rectangle in X has diameter at most 4Ram(n) + 3 in Γn X. For convenience, let dn
denote the distance in Γn X.
Let Q be a Ram(n)-thick flat rectangle in X. The set of the hyperplanes intersecting Q
can be naturally written as a disjoint union H t V where two hyperplanes both in H or
V are disjoint in Q.
Case 1: H or V contains n pairwise transverse hyperplanes. Say that H contains n
pairwise transverse hyperplanes H1 , . . . , Hn . A fortiori, for any hyperplane V ∈ V,
{V, H1 , . . . , Hn } is a collection of n + 1 pairwise transverse hyperplanes, hence dim V ≥
n; therefore, no hyperplane of V is n-combinatorially contracting. If x1 , x2 ∈ Q are
two vertices, let V1 , V2 ∈ V be two hyperplanes adjacent to x1 , x2 respectively, and let
y1 ∈ N (V1 ) ∩ N (H1 ), y2 ∈ N (V2 ) ∩ N (H1 ) be two vertices. Because V1 , V2 , H1 are not
n-combinatorially contracting, we deduce that
dn (x1 , x2 ) ≤ dn (x1 , y1 ) + dn (y1 , y2 ) + dn (y2 , x2 ) ≤ 3.
Therefore, diamn (Q) ≤ 3.
Case 2: Neither H nor V contains n pairwise transverse hyperplanes. Naturally, H (resp.
V) can be written as the disjoint union H = H1 t · · · t Hr (resp. V = V1 t · · · t Vs ) where
Hi (resp. Vi ) is a set of Ram(n) consecutive hyperplanes for all 1 ≤ i ≤ r (resp. for all
1 ≤ i ≤ s), and #Hr ≤ Ram(n) (resp. #Vs ≤ Ram(n)). Notice that, since Q is Ram(n)thick, necessarily r, s ≥ 2 hold. Now, for every 1 ≤ i ≤ r − 1 and 1 ≤ j ≤ s − 1, Hi
and Vj contain n pairwise disjoint hyperplanes; in particular, each of these hyperplanes
belongs to an (n, n)-grid of hyperplanes, and so is not n-combinatorially contracting.
Therefore, for every vertex a ∈ Q, there exists a vertex v(a) belonging to a hyperplane
V (a) ∈ V which is not n-combinatorially contracting, such that d(a, v(a)) ≤ 2Ram(n).
Let us fix a hyperplane H ∈ H which is not n-combinatorially contracting, and for every
vertex a ∈ Q, let w(a) be a vertex in V (a) ∩ H. Finally, for every x, y ∈ Q, we get
dn (x, y) ≤ dn (x, v(x)) + dn (v(x), w(x)) + dn (w(x), w(y)) + dn (w(y), v(y)) + dn (v(y), y)
≤ 2Ram(n) + 1 + 1 + 1 + 2Ram(n) = 4Ram(n) + 3
Thus, diamn (Q) ≤ 4Ram(n) + 3. 
Thus, looking at the induced action on the contracting graph, as a consequence of
Proposition 5.3 we obtain:
Corollary 5.4. Let G be a group acting geometrically on a CAT(0) cube complex X.
Then G is weakly hyperbolic relatively to the stabilizers of the non-contracting hyperplanes of X.
Recall that a group G is weakly hyperbolic relatively to a collection of subgroups H =
{H1 , . . . , Hn } if G acts by isometries on a graph Γ such that:
• Γ is hyperbolic,
• Γ contains finitely-many orbits of edges,
• each vertex-stabilizer is either finite or contains a finite-index subgroup conjugated
to some Hi ,
17

• any Hi stabilizes a vertex.
Proof of Corollary 5.4. Let Y be the graph obtained from X (1) by adding a vertex for
each non-contracting hyperplanes J and linking by an edge this vertex with any vertex
of N (J): this is the usual cone-off we will introduce in Definition 5.6, which is clearly
quasi-isometric to the cone-off of Definition 1.1. Thus, because Y is quasi-isometric
to the contracting graph Γ∞ X, we deduce from Proposition 5.3 that Y is hyperbolic.
Furthermore, the action G y X induces an action G y Y with finitely-many orbits of
edges, and the stabilizer of a vertex is either finite if it belongs to X or it is the stabilizer
of a non-contracting hyperplane. 
As an application of Corollary 5.4, we will be able to deduce a weak relative hyperbolicity
of right-angled Coxeter groups in Section 5.3.

5.2

Strong relative hyperbolicity

After considering the weak relative hyperbolicity, it is natural to focus on the strong
relative hyperbolicity. We recall the definition as introduced by Bowditch in [Bow12].
Definition 5.5. A finitely-generated group G is (strongly) hyperbolic relatively to a
collection of subgroups H = {H1 , . . . , Hn } if G acts by isometries on a graph Γ such
that:
• Γ is hyperbolic,
• Γ contains finitely-many orbits of edges,
• each vertex-stabilizer is either finite or contains a finite-index subgroup conjugated
to some Hi ,
• any Hi stabilizes a vertex,
• Γ is fine, ie., any edge belongs only to finitely-many simple loops (or cycle) of a
given length.
A subgroup conjugated to some Hi is peripheral. G is just said relatively hyperbolic if it
is relatively hyperbolic with respect to a finite collection of proper subgroups.
Thus, a natural question is: when is a cone-off of a CAT(0) cube complex fine? With
the definition of a cone-off we used in the previous sections, it is not difficult to notice
that essentially a cone-off will never be fine. So we first have to modify our definition:
Definition 5.6. Let X be a CAT(0) cube complex and Q a collection of subcomplexes.
The usual cone-off of X over Q is the graph obtained from X (1) by adding a vertex for
each subcomplex Q ∈ Q and linking it by an edge Q to each vertex belonging to Q.
Now, we are able to prove the following criterion.
Theorem 5.7. Let X be a uniformly locally finite CAT(0) cube complex and Y a usual
cone-off of X over a collection Q of combinatorially convex subcomplexes. If Q is locally
finite (ie., there exist only finitely-many subcomplexes of Q containing a given edge of X)
and if there exists a constant C ≥ 0 such that two subcomplexes of Q are both intersected
by at most C hyperplanes, then Y is fine. Conversely, if Q is not locally finite or if it
contains two subcomplexes both intersected by infinitely many hyperplanes, then Y is not
fine.

18

Proof. Suppose that Q is locally finite and that there exists a constant C ≥ 0 such
that two subcomplexes of Q are both intersected by at most C hyperplanes. Let e ∈ Y
be an edge and fix one of its endpoints a ∈ X. To a given cycle γ ⊂ Y of length n
containing e, we associate a loop γ̄ ⊂ X containing a in the following way: The cycle γ
passes through a sequence of cones C1 , . . . , Ck . For every 1 ≤ j ≤ k, let xj , yj ∈ X be
the two vertices of γ ∩ Cj ∩ X, and choose a combinatorial geodesic [xj , yj ] between xj
and yj in X. Also, for every 1 ≤ j ≤ k − 1, choose a combinatorial geodesic [yj , xj+1 ]
between yj and xj+1 in X. Finally, choose two combinatorial geodesics [a, x1 ] and [yn , a]
in X, respectively between a and x1 , and yn and a. Now we set
γ̄ = [a, x1 ] ∪ [x1 , y1 ] ∪ · · · ∪ [xk , yk ] ∪ [yk , a].
Fix some 1 ≤ j ≤ k. Notice that a hyperplane separating xj and yj must intersect
γ̄\[xj , yj ], ie., must separate either a and x1 , or a and y1 , or xi and yi for some 1 ≤ i 6=
j ≤ k, or yi and xi+1 for some 1 ≤ i ≤ k − 1. But a hyperplane separating xj and yj
and, for some 1 ≤ i ≤ k, xi and yi , necessarily intersects both Cj and Ci . Because there
exist at most C such hyperplanes, we deduce that
dX (xj , yj ) ≤ Ck + dX (a, x1 ) + dX (a, yk ) +

k−1
P

dX (yj , xj+1 )

j=1

≤ Cn + n = (C + 1)n
Therefore, we get
diamX (γ̄) ≤ dX (a, x1 ) +

k
P

dX (xj , yj ) +

j=1

k−1
P

dX (yj , xj+1 ) + dX (a, x1 )

j=1

≤ k(C + 1)n + n ≤ (C + 2)n2
We have proved that γ̄ is included into the ball B = BX (a, (C + 2)n2 ). Let Ḃ denote
the cone-off of this ball over {Q ∩ B | Q ∈ Q}, so that γ̄ ⊂ B implies γ ⊂ Ḃ. It is
worth noticing that Ḃ depends only on the edge e, the constant C and the integer n,
so Ḃ contains all the cycles of length n passing through e. Since Ḃ is finite by the local
finiteness of X and Q, we conclude that there exist only finitely many such cycles. This
proves that Y is fine.
Conversely, if Q is not locally finite, then there exist an edge e ∈ X and infinitelymany subcomplexes Q1 , Q2 , . . . ∈ Q containing e. For each i ≥ 1, e belongs to a cycle
of length three passing through the cone associated to Qi . Therefore, Y is not fine.
Now, suppose that Q contains two subcomplexes C1 , C2 both intersected by infinitelymany hyperplanes of X. Let p : X → C1 denote the combinatorial projection onto C2 .
According to Proposition 2.9, p(C2 ) is a geodesic subcomplex of C1 containing infinitelymany hyperplanes, say J1 , J2 , . . .. Choose a basepoint x ∈ p(C2 ) and, for every i ≥ 1,
fix a vertex xi ∈ J ⊂ p(C2 ) and a combinatorial geodesic [x, xi ] ⊂ p(C2 ). Because X
is locally finite, the sequence of geodesics ([x, xi ]) contains a subsequence converging
to some infinite ray r : [0, +∞) → p(C2 ). Because X is finite dimensional, we deduce
from the infinite Ramsey Theorem that the set of hyperplanes intersected by r must
contain an infinite sequence of pairwise disjoint hyperplanes, say V1 , V2 , . . .; notice that,
according to Proposition 2.9, the Vi ’s intersect C1 as well as C2 . For every n ≥ 1, let Cn
denote the cycle of combinatorially convex subcomplexes (N (V1 ), C1 , N (Vn ), C2 ). From
Corollary 2.17, we get a flat rectangle Dn bounded by C. Once again, because X is
locally finite, the sequence of subcomplexes (Dn ) contains a subsequence converging to
some subcomplexes D∞ . Notice that D∞ ∩ Ci is an infinite ray ri (with i = 1, 2) and
that D∞ ∩ N (Vi ) is a combinatorial geodesic γi between r1 and r2 whose length ` does
19

not depend on i (with i ≥ 1). For every n ≥ 2, let cn ⊂ Y denote the cycle passing
through γ1 , γn and the two cones over C1 and C2 . If e is any edge of γ1 , then cn is a
sequence of infinitely-many cycles with the same length containing e. Therefore, Y is
not fine. 
Now, we would like to combine Theorem 4.1 and Theorem 5.7 to deduce some criterion
of relative hyperbolicity. Although it seems to be difficult to state a general criterion,
this sketches a general approach to study the strong relative hyperbolicity of cubulable
groups. Roughly speaking:
Let G be a group acting geometrically on a CAT(0) cube complex X. If we are able
to find a G-equivariant collection of combinatorially convex subcomplexes C such that
any thick flat rectangle of X is included into a subcomplex of C, and two subcomplexes
of C do not fellow-travel, then, by looking at the induced action of G on the usual
cone-off of X over the collection Q, we should be able to deduce that G is strongly
hyperbolic relatively to {stab(Q) | Q ∈ Q}. In order to construct Q, a possibility could
be to start with the collection F 0 of the “combinatorial flats” of X, and then to define
inductively the sequence (F n ) by: if C1 , . . . , Ck denote the connected components of
the graph whose set of vertices is F n and
! whose edges link
!!two subcomplexes which
fellow-travel, we set F n+1 =

co

S

F

, . . . , co

F ∈C1

S

F

, where co(·) denotes the

F ∈Ck

combinatorial convex hull. Finally, because X is cocompact, the sequence (F n ) should
be eventually constant to some collection F ∞ , and this is our condidate for Q.
A similar idea can be found in [Cap09], and is successfully applied to Coxeter groups
[BHSC13, Theorem A.1]. In the next section, following the argument we have sketched
above, we will be able to give a direct proof of the second part of [BHSC13, Theorem
I], characterizing the strong relative hyperbolicity of right-angled Coxeter groups.

5.3

Application to right-angled Coxeter groups

Our plan is to apply the results established in the two previous sections to the class of
right-angled Coxeter groups. We begin with some basic definitions.
Definition 5.8. Let Γ be a simplicial graph. The right-angled Coxeter group C(Γ) is
defined by the presentation
hv ∈ V (Γ) | u2 = 1, [v, w] = 1, u ∈ V (Γ), (v, w) ∈ E(Γ)i,
where V (Γ) and E(Γ) denote respectively the sets of vertices and edges of Γ.
If Λ ≤ Γ is an induced subgraph (ie., two vertices of Λ are adjacent in Λ if and only if
they are adjacent in Γ), then the subgroup hΛi generated by the vertices of Λ is naturally
isomorphic to the right-angled Coxeter group C(Λ). For instance, if Λ is a clique of Γ
(ie., a complete subgraph) with n vertices, then hΛi is isomorphic to direct product of
n copies of Z2 : such a subgroup is referred to as a clique subgroup. Another interesting
class of subgroups is:
Definition 5.9. The join Λ1 ∗ Λ2 of two graphs Λ1 and Λ2 is the graph obtained from
the disjoint union Λ1 t Λ2 by adding an edge between any vertex of Λ1 and any vertex
of Λ2 . By extension, if a given simplicial graph Γ contains a subgraph Λ which splits
as a join of two non empty subgraphs, we say that the subgroup hΛi of the right-angled
Coxeter group C(Γ) is a join subgroup.
Given a simplicial graph Γ, the right-angled Coxeter group C(Γ) naturally acts on a
CAT(0) cube complex X(Γ) whose:
20

• vertices are the elements of C(Γ),
• edges link two vertices g, h if h = gv for some v ∈ V (Γ),
• n-cubes are generated by the vertices
{gvi1 · · · vir | 0 ≤ r ≤ n, 1 ≤ i1 < · · · < ir ≤ n},
where g ∈ C(Γ) and v1 , . . . , vn ∈ V (Γ) are pairwise adjacent.
Notice that the 1-skeleton of X(Γ) is the Cayley graph of C(Γ) with respect to the
generating set V (Γ). In particular, if Jv denotes the hyperplane dual to the edge (1, v)
where v ∈ V (Γ), then the hyperplanes of X(Γ) are precisely the translates of the Jv ’s.
The two following lemmas on the geometry of X(Γ) are elementary, and their proofs are
left to the reader. In the case of right-angled Artin groups, the analogous results have
been proved in [BC12, Section 3].
Lemma 5.10. If the hyperplanes gJa and hJb are transverse, then a 6= b and [a, b] = 1.
Lemma 5.11. The stabilizer of the hyperplane gJa is ghstar(a)ig −1 .
Recall that, given a graph Γ and a vertex v, the link of v, denoted by link(v), is defined
as the subgraph generated by the vertices of Γ adjacent to v; and its star, denoted by
star(v), is defined as the subgraph generated by {v} ∪ link(v).
We begin our study of the relative hyperbolicity of right-angled Coxeter groups by
determining precisely when a hyperplane of our cube complex is contracting.
Proposition 5.12. Let Γ be a finite graph and u ∈ V (Γ). The hyperplane Ju of X(Γ)
is contracting if and only if u does not belong to an induced square in Γ.
Proof. Suppose that the vertices u, v, w, x ∈ V (Γ) generate a square in Γ (with u and
x both adjacent to v and w). Then X(Γ) contains the following subcomplex for every
n ≥ 1:

It follows from Lemma 5.10 that the hyperplanes
({Ju , uJx , uxJu , . . . , (ux)n−1 uJx }, {Jv , vJw , vwJv , . . . , (vw)n−1 vJw })
define a (2n, 2n)-grid of hyperplanes. Thus, Ju is not contracting.
Conversely, suppose that Ju is not contracting. In particular, Ju belongs to a (L, L)grid of hyperplanes (H, V) satisfying L > n, where n is the maximal cardinality of a
clique subgroup of C(Γ). To fix the notation, say that H = {A1 , . . . , Ar , Ju , B1 , . . . , Bs }
and V = {V1 , . . . , VL }, with the convention that each of these hyperplanes separates
its two adjacent hyperplanes in the list it belongs. Without loss of generality, we may
21

suppose that s ≥ 1. Let Ju− (resp. Bs+ ) denote the half-space delimited by Ju (resp.
Bs ) not containing Bs (resp. Ju ) and V1− (resp. VL+ ) the half-space delimited by V1
(resp. VL ) not containing VL (resp. V1 ). According to Corollary 2.17, a disc diagram of
minimal complexity bounded by the cycle of subcomplexes (Ju− , VL+ , Bs+ , V1− ) will define
a subcomplex Q isomorphic to a rectangle. By translating Q by an element of hstar(u)i
if necessary, we may suppose without loss of generality that 1 is the corner of Q in
+
V1− ∩ Ju− . Let p1 · · · pi denote the corner of Q in Ju− ∩ V2L
and uq1 · · · qj the corner of Q
−
+
in V1 ∩ Bs , where p1 , . . . , pi , q1 , . . . , qj ∈ V (Γ).
Suppose by contradiction that u is adjacent to q1 , . . . , qj in Γ. Let B1 be the hyperplane
dual to the edge (uq1 · · · qk , uq1 · · · qk+1 ), ie., B1 = uq1 · · · qk Jqk+1 . Now, notice that Ju
and B1 are transverse if and only if qk · · · q1 uJu = Ju and Jqk+1 are transverse if and
only if u and qk+1 are adjacent vertices in Γ. On the other hand, Ju and B1 are disjoint
by assumption, so there exists 1 ≤ h ≤ s such that u and qh are not adjacent in Γ.
Moreover, we deduce from the definition of n that hp1 , . . . , pL i is an infinite subgroup,
so that there exist 1 ≤ ` < m ≤ L such that p` and pm are not adjacent in Γ.
Finally, noticing that u, q1 , . . . , qj are all adjacent to p1 , . . . , pi according to Lemma 5.10,
we conclude that u belongs to the induced square generated by u, qh , p` , pm in Γ. 
Thus, if (Γ) denote the set of the vertices of a graph Γ which belong to an induced
square, we obtain by combining Corollary 5.4, Proposition 5.12 and Lemma 5.11:
Proposition 5.13. Let Γ be a finite graph. The right-angled Coxeter group C(Γ) is
weakly hyperbolic relatively to {hstar(u)i | u ∈ (Γ)}.
In fact, during the proof of Proposition 5.12, we have shown:
Fact 5.14. Let Γ be a finite graph. If n denotes the maximal cardinality of a clique
subgroup of C(Γ), then any n-thick flat rectangle in X(Γ) is included into a subcomplex
X(Γ1 ∗ Γ2 ) ⊂ X(Γ), where Γ1 , Γ2 ⊂ Γ are not complete subgraphs.
Therefore, combined with Theorem 4.1, we deduce that right-angled Coxeter groups are
also weakly hyperbolic relatively to their join subgroups.
Proposition 5.15. Let Γ be a finite graph. The right-angled Coxeter group C(Γ) is
weakly hyperbolic relatively to {C(Γ1 ∗ Γ2 ) | Γ1 ∗ Γ2 ⊂ Γ where Γ1 , Γ2 are not complete}.
Remark 5.16. Proposition 5.13 and Proposition 5.15 should be compared with the
weak relative hyperbolicity established in [Bah14].
From now on, let us focus on the strong relative hyperbolicity. We begin with some
preliminary definitions.
Definition 5.17. Let Γ be a simplicial graph. A join Γ1 ∗ Γ2 in Γ is large if neither Γ1
nor Γ2 is a complete subgraph.
Definition 5.18. Let Γ be a finite graph. A join decomposition of Γ is a collection of
subgraphs (Γ1 , . . . , Γn ), with n ≥ 0, such that:
• any large join of Γ is included into Γi for some 1 ≤ i ≤ n,
• Γi ∩ Γj is complete for every 1 ≤ i < j ≤ n,
• for every vertex v, link(v) ∩ Γi not complete implies v ∈ Γi .
There exists at least one join decomposition: the trivial decomposition (Γ).

22

According to the following proposition, join decompositions lead to strong relative hyperbolicity.
Proposition 5.19. Let Γ be a finite graph and (Γ1 , . . . , Γn ) a join decomposition. Then
the right-angled Coxeter group C(Γ) is hyperbolic relatively to {C(Γ1 ), . . . , C(Γn )}.
We first need to prove the following lemma:
Lemma 5.20. Let X be a uniformly locally finite CAT(0) cube complex. For every
N ≥ 1, there exists a constant M ≥ 1 such that, for any collection H of at least M
hyperplanes, H contains two hyperplanes separated by at least N hyperplanes of X.
Proof. Let J1 , . . . , Jr be r hyperplanes such that, for every 1 ≤ i 6= j ≤ r, Ji and
Jj are not separated by N hyperplanes. For every 1 ≤ i ≤ r, let Ni denote the N neighborhood of Ji with respect to d∞ (see Section 6); according to [HW08, Corollary
3.5], Ni is combinatorially convex. Furthermore, our assumption implies that Ni ∩Nj 6= ∅
for every 1 ≤ i, j ≤ r, hence

r
T

Ni 6= ∅. Noticing that

i=1

d∞ ≤ d ≤ dim(X) · d∞ ,
we deduce that, if we fix some vertex c ∈

r
T

Ni , the hyperplanes J1 , . . . , Jr intersect

i=1

the ball B(c, N · dim(X)). On the other hand, X is uniformly locally finite, so the
cardinality of this ball is bounded above by a constant depending only on N , and a
fortiori the number of hyperplanes intersecting this ball is bounded above by a constant
κ(N ). Therefore, setting M = κ(N ) + 1 proves our lemma. 
Proof of Proposition 5.19. Let Y denote the cone-off of X(Γ) over the translates of
the X(Γi )’s. Combining Fact 5.14 and Theorem 4.1, we know that Y is hyperbolic. To
conclude, it is sufficient to deduce from Theorem 5.7 that Y is fine.
First, we will show that this collection of subcomplexes is locally finite. Indeed, if
an edge (g, gu) belongs to pairwise distinct subcomplexes h1 C(Λ1 ), h2 C(Λ2 ), . . ., where
T
T
Λi ∈ {Γ1 , . . . , Γn } for i ≥ 1, then u ∈
Λi and g ∈
hi C(Λi ). Since two different
i≥1

i≥1

cosets of a given subgroup are necessarily disjoint, we deduce that the Λi ’s have to
T
be pairwise distinct, otherwise the intersection
hi C(Λi ) would be empty. Because
i≥1

we have only n subgraphs in our join decomposition, this means that our collection
h1 C(Λ1 ), h2 C(Λ2 ), . . . is necessarily finite.
Now, we want to prove the separation property. Let N denote the maximal order of
a finite subgroup of C(Γ) and let M be the constant given by Lemma 5.20. Suppose
that there exist M hyperplanes intersecting both gX(Γi ) and hX(Γj ). By definition of
M , there exist two hyperplanes V1 , V2 , intersecting both gX(Γi ) and hX(Γj ), separated
by N hyperplanes. Therefore, since C = (V1 , gX(Γi ), V2 , hX(Γj )) is a cycle of four
combinatorially convex subcomplexes, we deduce from Corollary 2.17 that there exists
a flat rectangle D bounded by C. Let g1 , g2 , g3 , g4 denote the corners of D cyclically
ordered, with g1 , g2 ∈ hC(Γj ) and g3 , g4 ∈ gC(Γi ). Set w1 = g1−1 g2 and w2 = g1−1 g4 , so
that g2 = g1 w1 and g4 = g1 w2 ; notice that w1 and w2 commute, the product w1 w2 =
w2 w1 is reduced in C(Γ) and g3 = g1 w1 w2 = g1 w2 w1 . Let supp(wi ) denote the set of
letters (or equivalently, vertices) used to write the word wi (i = 1, 2). Because V1 and
V2 are separated by N hyperplanes, we deduce that hsupp(w1 )i has cardinality at least
N ; by the definition of N , this implies that hsupp(w1 )i is infinite, ie., supp(w1 ) contains
two non-adjacent vertices v1 , v2 .
From g1 , g2 ∈ hC(Γj ), we deduce that w1 = g1−1 g2 ∈ C(Γj ); from g3 , g4 ∈ gC(Γi ), we
deduce that w1 = g4−1 g3 ∈ C(Γi ). Therefore,
23

{v1 , v2 } ⊂ supp(w1 ) ⊂ Γi ∩ Γj .
This implies i = j.
Because w2 commute with w1 and because the product w1 w2 is reduced in C(Γ), any
vertex v ∈ supp(w2 ) satisfies
link(v) ∩ Γj ⊃ link(v) ∩ supp(w1 ) ⊃ {v1 , v2 }.
This implies v ∈ Γj . A fortiori, we get w2 ∈ Γj hence g4 = g1 w2 ∈ hC(Γj ) = hC(Γi ).
Thus, g4 ∈ gC(Γi ) ∩ hC(Γi ), so gC(Γi ) = hC(Γi ).
We have proved that if two subcomplexes of our collection are both intersected by N
hyperplanes then they are equal. This concludes the proof. 
Although finding a non trivial join decomposition implies that our right-angled Coxeter
group is relatively hyperbolic, a priori we do not know how to find such a decomposition,
and if so some decompositions are better than others. For instance, if a graph Γ contains
cut-vertices, then any choice of a set S of these cut-vertices produces a join decomposition
(Γ1 , . . . , Γr ), where, for every i 6= j, Γi ∩Γj is either empty or a single vertex of S; clearly,
the decomposition seems to be “finer” when S is the set of all the cut-points of Γ. This
is the motivation to introduce the following canonical join decomposition of a graph:
Definition 5.21. Let Γ be a finite graph. For every subgraph Λ ⊂ Γ, let cp(Λ) denote
the subgraph of Γ generated by Λ and the vertices v ∈ Γ such that link(v) ∩ Λ is
not complete. Now, define the collection of subgraphs Jn (Γ) of Γ by induction in the
following way:
• J0 (Γ) is the collection of all the large joins in Γ;
• if C1 , . . . , Ck denote the connected components of the graph whose set of vertices
is Jn (Γ) and whose edges link
with non-complete intersection, we
! two subgraphs!!
set Jn+1 (Γ) =

cp

S
Λ∈C1

S

Λ , . . . , cp

Λ

.

Λ∈Ck

Because Γ is finite, the sequence (Jn (Γ)) must be eventually constant to some collection
J∞ (Γ); see Example 5.26. By construction, it is clear that J∞ (Γ) is a join decomposition
of Γ. Furthermore, according to the next lemma, it may be thought of as the minimal
join decomposition.
Lemma 5.22. Let Γ be a finite graph and (Γ1 , . . . , Λm ) a join decomposition. For every
Λ ∈ J∞ (Γ), there exists some 1 ≤ i ≤ m such that Λ ⊂ Γi .
Proof. We prove the statement by induction on n. If n = 0, this is true by the definition
of a join decomposition. Now, suppose that the statement is true for Jn (Γ) and let
Λ ∈ Jn+1 (Γ). The subgraph Λ corresponds to a connected component C = {Λ1 , . . . , Λk }
of the graph whose set of vertices is Jn (Γ) and whose edges link two subgraphs with
non-complete intersection, ie., Λ = cp(Λ1 ∪ · · · ∪ Λk ). By our induction hypothesis, for
every 1 ≤ s ≤ k, there exists 1 ≤ is ≤ m such that Λs ⊂ Γis . Noticing that, for every
1 ≤ r < s ≤ k, Γik ∩ Γis contains the non-complete subgraph Λik ∩ Λis , we deduce that
ik = is ; let ι be this common value. We have
Λ = cp(Λ1 ∪ · · · ∪ Λk ) ⊂ cp(Γι ) = Γι .
This concludes the proof. 
Remark 5.23. In Definition 5.21, setting J0 (Γ) as the collection of the induced squares
in Γ does not affect J∞ (Γ). In practice, it may be simpler to determine the decomposition
J∞ (Γ) in this way. Moreover, it turns out that the subgraphs of J∞ (Γ) are precisely the
maximal subgraphs of Γ which belong to the class T introduced in [BHSC13].
24

Finally, we are able to characterize the strong relative hyperbolicity of right-angled
Coxeter groups, reproving the second part of [BHSC13, Theorem I].
Theorem 5.24. Let Γ be a finite graph. The right-angled Coxeter group C(Γ) is relatively hyperbolic if and only if J∞ (Γ) is not the trivial decomposition. If so, then C(Γ)
is hyperbolic relatively to the collection {C(Λ) | Λ ∈ J∞ (Γ)}.
Proof. Suppose that J∞ (Γ) is the trivial decomposition and that C(Γ) is hyperbolic
relatively to some collection of subgroups H. We will argue by induction on n that, for
any n ≥ 0 and Λ ∈ Jn (Γ), the subgroup C(Λ) is included into a peripheral subgroup.
Because any subgroup isomorphic to a direct product of two infinite groups has to be
included into a peripheral subgroup, the statement holds for n = 0. Now suppose
that this statement holds for some n ≥ 0, and let Λ ∈ Jn+1 (Γ). The subgraph Λ
corresponds to a connected component C = {Λ1 , . . . , Λk } of the graph whose set of
vertices is Jn (Γ) and whose edges link two subgraphs with non-complete intersection,
ie., Λ = cp(Λ1 ∪ · · · ∪ Λk ). By our induction hypothesis, for every 1 ≤ i ≤ k, the
subgroup C(Λi ) is included into some peripheral subgroup Hi . Notice that, for every
1 ≤ i < j ≤ k, Hi ∩ Hj contains C(Λi ) ∩ C(Λj ) = C(Λi ∩ Λj ), which is infinite, hence
Hi = Hj . Therefore, the C(Λi )’s are all included into the same peripheral subgroup H.
Then, for every vertex v ∈ cp(Λ1 ∪ · · · ∪ Λk ), we have
H ∩ H v ⊃ C(Λ1 ∪ · · · ∪ Λk ) ∩ C(Λ1 ∪ · · · ∪ Λk )v = C(Λ1 ∪ · · · ∪ Λk ),
so H ∩ H v is infinite, and this implies v ∈ H. Therefore,
C(Λ) = C(cp(Λ1 ∪ · · · ∪ Λk )) ⊂ H.
This concludes the proof of our claim. Finally, because J∞ (Γ) = (C(Γ)), we deduce
that some peripheral subgroup of H is not proper. Consequently, C(Γ) is not relatively
hyperbolic.
The converse is a consequence of Proposition 5.19. 
Remark 5.25. It is worth noticing that the previous proof shows that the collection
of subgroups {C(Λ) | Λ ∈ J∞ (Γ)} is a minimal collection of peripheral subgroups, ie.,
if C(Γ) hyperbolic relatively to H, then, for every Λ ∈ J∞ (Γ), there exists H ∈ H such
that C(Λ) ⊂ H.
Example 5.26. The figure below gives an example of a graph Γ and its canonical join
decomposition J∞ (Γ).

In particular, we deduce from Theorem 5.24 that the associated right-angled Coxeter
group C(Γ) is relatively hyperbolic.

25

6

Hyperbolicity with respect to the `∞ metric

Recall that the distance d∞ on a CAT(0) cube complex X is the extension on the whole
cube complex of the `∞ -norm defined on each cube. Alternatively, the restriction of d∞
over X (0) corresponds to the graph metric associated to the cone-off of X over its cubes.
Because, loosely speaking, d∞ “kills” the dimension of X, it may be expected, following
Theorem 3.3, that (X, d∞ ) is hyperbolic precisely when its grids of hyperplanes cannot
be too large. This is precisely what we prove.
Theorem 6.1. Let X be a CAT(0) cube complex. Then (X, d∞ ) is hyperbolic if and
only if the grids of hyperplanes in X are uniformly thin.
We begin by proving a preliminary lemma.
Lemma 6.2. Let X be a CAT(0) cube complex and x, y, z, z 0 ∈ X four pairwise distinct
vertices. Let m = m(x, y, z) and m0 = m(x, y, z 0 ) be the associated median vertices.
Then any hyperplane separating m and m0 separates z and z 0 as well. In particular,
d∞ (m, m0 ) ≤ d∞ (z, z 0 ).
Proof. Fix eight combinatorial geodesics [x, m], [y, m], [z, m], [x, m0 ], [y, m0 ], [z, m0 ],
[m, m0 ], [z, z 0 ]. We get a bigon B = {[x, m] ∪ [m, y], [x, m0 ] ∪ [m0 , y]} and a 4-gon
P = {[m, m0 ], [m0 , z 0 ], [z 0 , z], [z, m]}. Given a hyperplane J separating m and m0 , in
B necessarily either J intersects [x, m0 ] and [y, m] or it intersects [x, m] and [y, m0 ]. Say
we are in the first case, the second one being completely similar. In particular, J cannot
intersect neither [m, z] nor [m0 , z 0 ] since otherwise J would intersect the combinatorial
geodesic [y, m] ∪ [m, z] or [x, m0 ] ∪ [m0 , z 0 ] twice. On the other hand, J has to intersect
the 4-gon P since it separates m and m0 , so we conclude that J must separate z and z 0 .

Proof of Theorem 6.1. Suppose that the grids of hyperplanes in X are all C-thin for
some C ≥ 1. According to Lemma 4.2, it is sufficient to prove that the bigons in X are
all (C + 3)-thin with respect to d∞ .
Let {γ1 , γ2 } ⊂ X be a bigon with endpoints x, y; for convenience, let ` = dX (x, y).
Let us fix some vertex z ∈ γ1 , and, for any 0 ≤ t ≤ `, let z(t) denote the vertex
of γ2 defined by dX (x, z(t)) = t. We may suppose that d∞ (x, z), d∞ (z, y) ≥ C + 2,
otherwise there is nothing to prove. Finally, let m1 (t) and m2 (t) be the median points
associated to {x, z, z(t)} and {y, z, z(t)} respectively, and let F denote the function
t 7→ d∞ (z, m1 (t)) − d∞ (z, m2 (t)).

Notice that, using Lemma 6.2, we get
d∞ (m2 (1), z = m(x, y, z)) ≤ d∞ (x, z(1)) = 1.
Similarly, d∞ (m1 (` − 1), z) ≤ 1. Futhermore, we have
d∞ (z, m1 (1)) ≥ d∞ (z, x) − d∞ (x, m1 (1)) ≥ C + 2 − 1 = C + 1.
26

Similarly, d∞ (z, m2 (` − 1)) ≥ C + 1. Therefore, we conclude that F (1) ≥ C and
F (` − 1) ≤ −C. On the other hand, for all 1 ≤ t ≤ d(x, y) − 1, we have
|F (t + 1) − F (t)| = |d∞ (z, m1 (t + 1)) − d∞ (z, m2 (t + 1)) − d∞ (z, m1 (t)) + d∞ (z, m2 (t))|
≤ |d∞ (z, m1 (t + 1)) − d∞ (z, m1 (t))| + |d∞ (z, m2 (t + 1)) − d∞ (z, m2 (t))|
≤ d∞ (m1 (t), m1 (t + 1)) + d∞ (m2 (t), m2 (t + 1))
≤ d∞ (z(t), z(t + 1)) + d∞ (z(t), z(t + 1)) = 2
where the last inequality is justified by Lemma 6.2. Consequently, there exists some
1 ≤ t0 ≤ d(x, y) such that |F (t0 )| ≤ 3. That is to say, if V1 , . . . , Vr (resp. H1 , . . . , Hs ) is
a maximal collection of pairwise disjoint hyperplanes separating z and m1 (t0 ) (resp. z
and m2 (t0 )), ie., with r = d∞ (z, m1 (t0 )) (resp. s = d∞ (z, m2 (t0 ))), then |r − s| ≤ 3.
Applying Fact 4.4, we notice that z is the median point m(z, m1 (t0 ), m2 (t0 )), and similarly z(t0 ) is the median point m(z(t0 ), m1 (t0 ), m2 (t0 )). Thus, no hyperplane can separate z or z(t0 ) from {m1 (t0 ), m2 (t0 )}; we deduce that the Vi ’s separate {z, m2 (t0 )} and
{m1 (t0 ), z(t0 )}, and similarly the Hj ’s separate {z, m1 (t0 )} and {m2 (t0 ), z(t0 )}. It follows that Vi and Hj are transverse for any 1 ≤ i ≤ r and 1 ≤ j ≤ s, and our hypothesis
implies that min(r, s) ≤ 3. Finally, we conclude that
d∞ (z, γ2 ) ≤ d∞ (z, z(t0 )) = max(r, s) ≤ C + 3,
ie., the bigon {γ1 , γ2 } is (C + 3)-thin.
Conversely, suppose that (X, d∞ ) is δ-hyperbolic. Let (H, V) be a grid of hyperplanes
in X. We want to prove that min(#H, #V) ≤ 4δ + 2. Of course, if #V ≤ 4δ + 3, there
is nothing to prove, so we may suppose without loss of generality that #V > 4δ + 3.
For convenience, let H = {H1 , . . . , Hr } (resp. V = {V1 , . . . , Vs }) such that Hi separates
Hi−1 and Hi+1 for all 1 ≤ i ≤ r (resp. Vj separates Vj−1 and Vj+1 for all 1 ≤ j ≤ s); let
H1− and Hr+ (resp. V1− and Vs+ ) be the two disjoint half-spaces delimited respectively
by H1 and Hr (resp. V1 and Vs ).
Let x ∈ Hr+ ∩ V1− , y ∈ Vs+ ∩ Hr+ , z ∈ Vs+ ∩ H1− and w ∈ V1− ∩ H1− be four vertices,
and fix five d∞ -geodesics γ(x, y), γ(y, z), γ(z, w), γ(w, x) and γ(y, w). According to
Proposition 2.5, we may suppose without loss of generality that γ(x, y) ⊂ Hr+ , γ(y, z) ⊂
Vs+ , γ(z, w) ⊂ H1− and γ(w, x) ⊂ V1− . Let p ∈ γ(x, y) be a vertex satisfying |d∞ (x, p) −
d∞ (x, y)/2| ≤ 1/2. Notice that
d∞ (p, γ(x, w)) ≥

s
2

− 2 > δ,

since V1 , . . . , Vbs/2c−1 separates p and γ(x, w). Thus, because the geodesic triangle
γ(x, y) ∪ γ(y, w) ∪ γ(w, x) is δ-thin by assumption, there exists a vertex p0 ∈ γ(y, w)
such that d∞ (p, p0 ) ≤ δ. Similarly, noticing that
d∞ (p0 , γ(y, z)) ≥ d∞ (p, γ(y, z)) − d∞ (p, p0 ) ≥

s
2

− 1 − δ > δ,

since Vbs/2c+1 , . . . , Vs separates p and γ(y, z), we deduce that there exists a vertex p00 ∈
γ(z, w) satisfying d∞ (p0 , p00 ) ≤ δ, because the geodesic triangle γ(w, y) ∪ γ(y, z) ∪ γ(z, w)
is δ-thin. On the other hand, H1 , . . . , Hr separates p and γ(w, z) (by d∞ -convexity of
half-spaces), hence
r ≤ d∞ (p, γ(z, w)) ≤ d∞ (p, p00 ) ≤ d∞ (p, p0 ) + d∞ (p0 , p00 ) ≤ 2δ.
We have proved that either #V ≤ 4δ +2 or #H ≤ 2δ. A fortiori, min(#V, #H) ≤ 4δ +2.

27

7

Acylindrical actions

The main result of this section is the following criterion, establishing the equivalence
between several acylindrical properties of an action. Notice however that the acylindrical property we consider is weaker from acylindrical actions as defined in [Osi13] (see
Definition 8.17), because of a lack of uniformity on the constants.
Theorem 7.1. Let G be a group acting on a complete CAT(0) cube complex X. Suppose
(X, d∞ ) hyperbolic. The following statements are equivalent:
(i) for every d ≥ 0, there exists R ≥ 0 such that, for every vertices x, y ∈ X,
d∞ (x, y) ≥ R ⇒ #{g ∈ G | d∞ (x, gx), d∞ (y, gy) ≤ d} < +∞;
(ii) there exists R ≥ 0 such that, for every vertices x, y ∈ X,
d∞ (x, y) ≥ R ⇒ #{g ∈ G | gx = x, gy = y} < +∞;
(iii) there exists R ≥ 0 such that, for any hyperplanes J1 , J2 separated by at least R
pairwise disjoint hyperplanes, stab(J1 ) ∩ stab(J2 ) is finite.
The following lemma will be needed to prove this theorem:
Lemma 7.2. Let G be a group acting on a complete CAT(0) cube complex X. If
G y (X, d∞ ) has a bounded orbit, then G stabilizes a cube.
It is a consequence of [Pol93, Theorem A], stating that a bounded Helly graph without
infinite clique contains a clique invariant by any automorphism, since it is proved in
[BvdV91, Proposition 2.6] that the cone-off associated to (X, d∞ ) is a Helly graph.
Proof of Theorem 7.1. The implication (i) ⇒ (ii) is clear.
Now, we want to prove (ii) ⇒ (iii). According to Theorem 6.1, there exists a constant
C ≥ 1 such that any grid of hyperplanes in X is C-thin; let R denote the constant given
by (ii). Let J1 , J2 be two hyperplanes separated by at least max(C + 1, R) pairwise
disjoint hyperplanes, say V1 , . . . , Vr . Suppose that K = stab(J1 )∩stab(J2 ) is infinite and
that the action K y J1 has unbounded orbits (with respect to d∞ ). Thus, if x ∈ N (J1 )
and y ∈ N (J2 ) are two vertices minimizing the combinatorial distance between N (J1 )
and N (J2 ), there exists k ∈ K such that d∞ (x, kx) > C, ie., there exist C + 1 pairwise
disjoint hyperplanes H1 , . . . , HC+1 separating x and kx. A fortiori, the vertices kx
and ky also minimize the combinatorial distance between N (J1 ) and N (J2 ), so that,
according to Proposition 2.7, the hyperplanes separating x and y, and kx and ky, are the
same: namely, the hyperplanes separating J1 and J2 . As a consequence, any hyperplane
separating x and kx necessarily separates y and ky, because otherwise it would separate
x and y or kx and ky, but this is impossible; in particular, the Hi ’s separate y and
ky. Therefore, the Hi ’s and the Vj ’s define a grid of hyperplanes which is not C-thin,
contradicting the definition of C. Consequently, the action K y J1 has a bounded
orbit (with respect to d∞ ); of course, the same statement holds for K y J2 . Now,
Lemma 7.2 implies that K contains a finite-index subgroup K0 fixing pointwise two
cubes Q1 ⊂ N (J1 ) and Q2 ⊂ N (J2 ): this contradicts (ii) since K0 is infinite.
Finally, we want to prove (iii) ⇒ (i). Let R ≥ 1 denote the constant given by (iii). Let
d ≥ 1 and let x, y ∈ X be two vertices satisfying d∞ (x, y) ≥ R + 2d + 2. Our aim is to
prove that the set
F = {g ∈ G | d∞ (x, gx), d∞ (y, gy) ≤ d}
28

is finite. Let W be a collection of R + 2d + 2 pairwise disjoint hyperplanes separating x
and y. We claim that, for every g ∈ F , the images by g of all but at most 2d hyperplanes
of W separate x and y. Indeed, obviously any hyperplane of gW must separate gx and
gy, so that if such a hyperplane does not separate x and y, necessarily it has to separate
either x and gx or y and gy. Therefore, the hypothesis d∞ (x, gx), d∞ (y, gy) ≤ d proves
our claim.
Thus, if we let L denote the set of functions from a subset of W whose cardinality
is at least R + 2 to the set of hyperplanes separating x and y, then any element of
F naturally induces an element of L. Therefore, because L is finite, if F is infinite
there exist infinitely many elements g0 , g1 , g2 , . . . ∈ F inducing the same element of
L; in particular, we get infinitely many elements g0−1 g1 , g0−1 g2 , . . . ∈ G stabilizing each
hyperplane of a subfamily W0 ⊂ W of cardinality at least R + 2. Finally, W0 contains
two hyperplanes J1 , J2 separated by at least R pairwise disjoint hyperplanes, such that
stab(J1 ) ∩ stab(J2 ) is infinite, contradicting (iii). Consequently, (iii) implies that F is
necessarily finite. 

8

Acylindrical hyperbolicity of infinitely-presented groups

8.1

Small cancellation polygonal complexes

In the following, a polygonal complex X will be a two-dimensional CW-complex whose
cells, which are polygons, embed. In this context, a disc diagram is a continuous combinatorial map D → X, where D is a finite contractible polygonal complex with a fixed
topological embedding into S2 ; notice that D may be non-degenerate, ie., homeomorphic
to a disc, or may be degenerate. In particular, the complement of D in S2 is a 2-cell,
whose attaching map will be referred to as the boundary path ∂D → X of D → X; it is
a combinatorial path.
A disc diagram D → X is reduced if, for any polygons P, Q ⊂ D meeting along
an edge e, whenever we think of the boundaries of P, Q as two combinatorial paths
∂P, ∂Q → X with e as the first edge, then these boundaries are different.
Given a combinatorial closed path P → X, we say that a disc diagram D → X is
bounded by P → X if there exists an isomorphism P → ∂D such that the following
diagram is commutative:
∂D
7/ X
O
P
According to a classical argument due to van Kampen and Lyndon, there exists a reduced
disc diagram bounded by a given combinatorial closed path if and only if this path is
null-homotopic.
Definition 8.1. Let X be a polygonal complex. A piece in X is a combinatorial path
p included into the intersection of two distinct polygons; let |p| denote its length. We
say that X satisfies
• the condition C 0 (λ) if any piece p between two polygons P, Q satisfies |p| < λ|P |;
• the condition C(n) if the boundary of a polygon cannot be covered by k pieces
with k < n;
• the condition T (n) if the length of a cycle in the link of a vertex of X is either two
or at least n.
Notice that the condition C 0 (1/n) implies the condition C(n).
29

The basic idea is that a reduced disc diagram D → X in a small cancellation polygonal
complex X necessarily contains a polygon with a large part of its boundary in ∂D. This
is the meaning of the next result.
Definition 8.2. Let X be a polygonal complex and D → X a disc diagram. A spur
in D is a vertex of degree one. An i-shell is a polygon P of D whose boundary can be
written as the concatenation of an outer path ∂out P and an inner path ∂inn P such that
∂out P = ∂P ∩ ∂D, ∂inn P ∩ ∂D does not contain edges and ∂inn P is covered by at most
i pieces.
It is worth noticing that a non-degenerate disc diagram containing at least one polygon
cannot contain a spur.
Theorem 8.3. [MW02, Theorem 9.4] Let X be a C(4) − T (4) polygonal complex and
D → X a reduced disc diagram. Then either D contains two spurs and/or i-shells with
i ≤ 2, or D is a single vertex or a single 2-cell.
From now on, we will suppose that any polygon of a polygonal complex has an even
number of sides. This assumption is not really restrictive since any polygonal complex
satisfies it up to a subdivision of its edges; nevertheless, it is worth noticing that this
process does not disturb the small cancellation condition. We want to sketch how Wise
associates a CAT(0) cube complex to some small cancellation polygon complexes in
[Wis04].
Definition 8.4. Let X be a polygonal complex. A hypergraph Λ is an equivalence class
of edges with respect to the relation: two edges e, e0 are equivalent if there exists a
sequence of edges e0 = e, e1 , . . . , en−1 , en = e0 such that, for every 0 ≤ i ≤ n − 1, ei
and ei+1 are opposite edges in a polygon of X. Alternatively, Λ may be thought of as a
graph included into X whose vertices are the midpoints of the edges which belong to the
associated equivalence class and whose edges are segments between two opposite cells
in a polygon. The hypercarrier of Λ, denoted by Y (Λ), is the union of all the polygons
of X intersected by Λ.
In [Wis04], Wise shows that the hypergraphs and their hypercarriers have good properties. In particular, they embed into the associated polygonal complex, hypercarriers are
combinatorially convex, and hypergraphs disconnect the polygonal complex into exactly
two connected components. The latter property allows us to define a halfcarrier of a
hypergraph Λ as the union of the hypercarrier Y (Λ) of Λ with a connected component
of the complement of Λ.
The following description of hypercarriers is especially useful:
Proposition 8.5. [Wis04, Lemma 3.6] Let X be a C(4) polygonal complex and Λ a
hypergraph. Then,
• the boundary of each polygon of Y (Λ) contains a pair of disjoint non-trivial arcs,
called isolated rails, and the interior of each isolated rail is disjoint from any other
polygon of Y (Λ);
• for each dual edge there is a tree, called a rung tree, such that distinct dual edges
have disjoint rung trees;
• the 1-skeleton of Y (Λ) is the union of rung trees and isolated rails such that rung
trees corresponding to adjacent vertices of Λ are attached together by a pair of
isolated rails corresponding to the edge of Λ connecting these vertices.
Let us mention a consequence of this result which will be useful in the next section.
30

Corollary 8.6. Let X be a C(4) polygonal complex, R ⊂ X a polygon, a and b two
opposite edges in R, and f * R an edge adjacent to R. Let Y denote the hypercarrier
of the hypergraph dual to a and b. If f ⊂ Y , then there exists a polygon S ⊂ X such
that either a and f , or b and f , are included into ∂S.
Proof. If f ⊂ Y , a fortiori there exists a polygon S containing f . It follows directly
from Proposition 8.5 that the intersection between two polygons in Y is either empty or
contains an edge dual to the associated hypergraph. Therefore, R ∩ S 6= ∅ implies that
∂S contains a or b. 
As already mentionned, the hypergraphs of a polygonal complex X disconnect the complex into exactly two connected components. In fact, the set of hypergraphs endows
X with a structure of a wallspace, so that there exists a CAT(0) cube complex C(X)
associated to X, which is constructed by cubulating this wallspace. See [Wis04, Section
5] for more information. In particular, we have a nice description of the maximal cubes
in C(X).
Definition 8.7. An isolated edge in X, ie., an edge which does not belong to any
polygon, gives rise to an edge in C(X), which we call an edge-cube. A polygon with 2n
sides in X gives rise to an n-cube in C(X), which we call a cell-cube.
Proposition 8.8. [Wis04, Lemma 9.4] Let X be a C(4) − T (4) polygonal complex and
C(X) its associated CAT(0) cube complex. Then any cube in C(X) is contained into an
edge-cube or into a cell-cube.
Corollary 8.9. The CAT(0) cube complex associated to a C(4)−T (4) polygonal complex
is complete, ie., it does not contain any infinite increasing sequence of cubes.

8.2

A fixed point theorem

In order to link the stabilisers in a small cancellation polygonal complex and in its
associated cube complex, we need the following result:
Theorem 8.10. Let G be a group acting on a C 0 (1/4) − T (4) polygonal complex X
and let C(X) denote the associated CAT(0) cube complex. There exists a G-equivariant
projection p : C(X) → X. Moreover, if two points x, y ∈ C(X) are separated by at
least R + 2 pairwise disjoint hyperplanes, then p(x) and p(y) are separated by at least R
pairwise disjoint hypergraphs.
We begin by proving several preliminary lemmas. From now on, we fix a C 0 (1/4) − T (4)
polygonal complex X.
Lemma 8.11. The intersection between two polygons in X is connected.
Proof. Suppose by contradiction that X contains two polygons P1 , P2 whose intersection is not connected. In particular, there exist two combinatorial paths γ1 ⊂ P1 and
γ2 ⊂ P2 with the same endpoints, and such that the concatenation γ = γ1 ∪ γ2 defines a
simple loop in X. Let D → X be a reduced non-degenerate disc diagram bounded by γ.
If D is a single cell, then its image in X defines a polygon whose boundary is included
into ∂P1 ∪ ∂P2 : this clearly contradicts the condition C 0 (1/4). According to Theorem
8.3, D must contain a shell; let R denote its image in X. The outer path of R is covered
by at most two pieces, namely ∂P1 and ∂P2 , and its inner path is covered by at most
two pieces by definition, so ∂R is covered by at most four pieces: this contradicts the
condition C 0 (1/4). 
The previous lemma will be used frequently in what follows, without any further reference.
31

Lemma 8.12. For any collection of pairwise intersecting polygons P1 , . . . , Pn , the intersection

n
T

Pi is non-empty.

i=1

Proof. We argue by induction on n. For n = 1, 2, there is nothing to prove. For
n = 3, suppose by contradiction that there exist three pairwise intersecting polygons
P1 , P2 , P3 such that P1 ∩ P2 ∩ P3 = ∅. In particular, there exist three combinatorial
paths γ1 ⊂ ∂P1 , γ2 ⊂ ∂P2 , and γ3 ⊂ ∂P3 , such that γi ∩ γi+1 is a single vertex {vi,i+1 }
for every i ∈ Z3 . Thus, the concatenation γ = γ1 ∪ γ2 ∪ γ3 defines a simple loop in X,
and bounds a reduced non-degenerate disc diagram D → X. If D is a single cell, then
its image in X is a polygon whose boundary is covered by three pieces, namely ∂P1 ,
∂P2 and ∂P3 : this contradicts the condition C 0 (1/4). We deduce from Theorem 8.3 that
D contains two shells; let R1 , R2 denote their images in X. Because, by definition, the
inner path of ∂R1 is covered by at most two pieces, the condition C’(1/4) implies that
the outher path of ∂R1 must be covered by at least three pieces: therefore, the outer
path of ∂R1 must contain some γi and intersect the two other subpaths of γ along at
least one edge. Of course, the same statement holds for R2 . But this contradicts the fact
that the intersection between the outer paths of R1 and R2 contains no edges. Thus,
necessarily P1 ∩ P2 ∩ P3 6= ∅.
Now, let n ≥ 4 be some integer and suppose that our lemma holds for every k < n. By
our induction hypothesis, for every 1 ≤ k ≤ n, the intersection
Ik = P1 ∩ · · · ∩ Pk−1 ∩ Pk+1 ∩ · · · ∩ Pn
n
T

Pi is empty, ie., Ii ∩ Ij = ∅ for every
i=1
∂P1 whose complement contains ∂Pk ∩∂P1 ;

is non-empty. Suppose by contradiction that

i 6= j. For every k ≥ 2, Ik is a subsegment in
notice furthermore that ∂Pk ∩ ∂P1 contains Ij for every j 6= k. Therefore, ∂P1 is covered
by ∂P2 , ∂P3 , and ∂P4 , contradicting the condition C’(1/4). We conclude that

n
T

Pi is

i=1

non-empty. 
Lemma 8.13. Two disjoint polygons are separated by a hypergraph.
Proof. Let us say that a halfcarrier contains strictly a polygon if this polygon does not
belong to the hypercarrier of the associated hypergraph. For any polygon P , let K(P )
denote the intersection of all the halfcarriers containing strictly P and star(P ) the union
of all the closed cells intersecting P . To prove our lemma, it is sufficient to show that
K(P ) = star(P ) for every polygon P ⊂ X.
The inclusion star(P ) ⊂ K(P ) is clear since star(P ) is included into any halfcarrier
containing strictly P . Conversely, notice that K(P ) is connected, as an intersection
of combinatorially convex subcomplexes; therefore, to prove that K(P ) ⊂ star(P ), it is
sufficient to show that, for any edge f * star(P ) adjacent to star(P ), we have f * K(P ),
ie., there exists a hypergraph Λ separating f and P such that f * Y (Λ).
Case 1: there exists an edge e adjacent to both P and f . It follows from Proposition 8.5
that the hypergraph Λ dual to e does not intersect f nor P : Λ separates f and P . Let
R denote a polygon in Y (Λ) containing e and e0 ⊂ R the edge parallel to e. According
to Corollary 8.6, if f ⊂ Y (Λ) there exists a polygon S containing e and f , but this
implies f ⊂ star(P ), or containing e0 and f , so that ∂S covers at least one half of ∂R,
contradicting the condition C’(1/4). Therefore, f * Y (Λ).
Case 2: there exists a polygon R0 adjacent to both P and f . Let R be a cell containing
f and such that the length of the intersection |∂R0 ∩ ∂R| is maximal. Let γ1 , γ2 denote
the two subsegments of ∂R0 between ∂P ∩ ∂R0 and ∂R0 ∩ ∂R. Notice that f * star(P )
implies that the lengths of γ1 and γ2 are positive.
32

Suppose by contradiction that no hypergraph intersects both γ1 and γ2 . Let γ1 denote
the subsegment of R0 opposite to γ1 . Our assumption implies that the intersection
γ1 ∩ γ2 contains no edges; in particular, γ1 is included into ∂R or ∂P , say γ1 ⊂ ∂R.
Then ∂R covers at least one half of ∂R0 , contradicting the condition C 0 (1/4). Therefore,
there exists a hypergraph Λ intersecting both γ1 and γ2 .
If follows from Proposition 8.5 that Λ does not intersect P nor f , ie., Λ separates P and
f . Let e1 (resp. e2 ) denote the edge of γ1 ⊂ R0 (resp. γ2 ⊂ R0 ) dual to Λ. If f ⊂ Y (Λ),
Corollary 8.6 implies that there exists a polygon S containing either f and e1 or f and
e2 . If S contains f and e1 , then the three polygons R0 , R and S define a disc diagram
in which the vertex f ∩ R0 is interior of degree three, contradicting the condition T(4).
If S contains f and e2 , then this contradicts the choice of R since
length(∂R0 ∩ ∂S) > length(∂R0 ∩ ∂R1 ).
Therefore, f * Y (Λ). This concludes the proof. 
Proof of Theorem 8.10. Let x ∈ C(X). If x belongs to an edge-cube Q, there exists
an isolated edge e in X with endpoints y, z ∈ X (0) such that Q is an edge between the
principal ultrafilters v(y) and v(z) respectively defined by y and z. Now, we define p(x)
as the only point of e such that d(x, v(y)) = d(p(x), y) and d(x, v(z)) = d(p(x), z).
From now on, suppose that x does not belong to any edge-cube. Thus, according to
Proposition 8.8, there is a family of polygons {Pi | i ∈ I} naturally associated to the
collection of the maximal cubes {Qi | i ∈ I} containing x. It follows from Lemma 8.13
that two polygons Pi and Pj intersect if and only if the associated cubes Qi and Qj
intersect. Therefore, {Pi | i ∈ I} is a collection of pairwise intersecting polygons in
X. Because the 0-skeleton of a polygon is finite, necessarily there exists a finite subset
J ⊂ I satisfying
T
i∈I

T

Pi =

Pj .

j∈J

It follows from Lemma 8.12 that this intersection M (x) ⊂ X is non-empty. If M (x) is
a polygon, we define p(x) as the center of this polygon. Otherwise, M (x) is a segment,
and we define p(x) as its midpoint.
The G-equivariance of p : C(X) → X follows by construction.
Finally, suppose that two points x, y ∈ C(X) are separated by at least R + 2 pairwise
disjoint hyperplanes. Let Q(x) (resp. Q(y)) be a maximal cube containing x (resp. y)
and let P (x) (resp. P (y)) denote the corresponding edge or polygon in X. Because
the hyperplanes intersecting a cube are pairwise transverse, we deduce that Q(x) and
Q(y) are separated by at least R pairwise disjoint hyperplanes; a fortiori, P (x) and
P (y) are separated by at least R pairwise disjoint hypergraphs. Since p(x) ∈ P (x) and
p(y) ∈ P (y), the conclusion follows. 
As a consquence of Theorem 8.10, we are able to deduce the following fixed point
theorem:
Corollary 8.14. Let G be a group acting by isometries on a C 0 (1/4) − T (4) polygonal
complex X. If G has a bounded orbit, then it has a global fixed point.
33

Proof. If G y X has a bounded orbit, then the associated action G y C(X) on
the CAT(0) cube complex C(X) has also a bounded orbit. Therefore, C(X) contains
a global fixed point x ∈ C(X). Now, if p : C(X) → X denotes the G-equivariant
projection given by Theorem 8.10, then G fixes p(x). 
A second fixed point theorem is:
Theorem 8.15. Let G be a finitely-generated group acting by isometries on a C 0 (1/4) −
T (4) polygonal complex X. If any element of G has a bounded orbit, then G has a global
fixed point.
Essentially, this result will be a consequence of the following statement.
Proposition 8.16. Let G be a finitely-generated group acting on a CAT(0) cube complex
C. If G y (C, d∞ ) contains an unbounded orbit, then there exists a half-space D and
an element g ∈ G such that gD ( D.
A similar statement was proved by Sageev during the proof of [Sag95, Theorem 5.1].
We follow his argument below.
Proof. Let S = {s1 , . . . , sr } be a finite generating set of G satisfying S −1 = S and
v ∈ C a base vertex. By assumption, the orbit G · v is unbounded with respect to d∞ ,
so there exists some g ∈ G such that d∞ (v, gv) ≥ 3M where M =

r
P

d(v, si v). Let us

i=1

write g as a product of elements of S: g = si(1) · · · si(m) . If, for every 1 ≤ i ≤ r, we fix a
combinatorial geodesic γi between v and si v, then the concatenation
α = γi(1) ∪ si(1) γi(2) ∪ si(1) si(2) γi(3) ∪ · · · ∪ si(1) · · · si(m−1) γi(m)
defines a combinatorial path between v and gv. Now, because d∞ (v, gv) ≥ 3M , there
exists a collection of pairwise disjoint hyperplanes V1 , . . . , VL separating v and gv, with
L ≥ 3M . Of course, any of these hyperplanes separates α, so any Vi is a translate of
some Hjk . Because the number of Hjk ’s is at most M and L ≥ 3M , we deduce that there
exist three Vi ’s which belong to the orbit of the same Hjk .
We have proved that there exist a hyperplane J and two elements g, h ∈ G such that
J, gJ, hJ are three pairwise disjoint hyperplanes separating v and gv. For convenience,
say that gJ separates J and hJ, and let D be the half-space delimited by J which
contains hJ. If gD ⊂ D or hD ⊂ D, we have done. Suppose that this not the case, ie.,
X\D ⊂ gD and X\D ⊂ hD or equivalently D ⊃ X\gD and D ⊃ X\hD. Then
hgD = X\h(X\gD) ⊂ X\hD ⊂ D.
This concludes the proof. 
Proof of Theorem 8.15. Let C(X) denote the CAT(0) cube complex associated to X.
Because any element of G has a bounded orbit with respect to the action G y X, the
same assertion holds with respect to G y (C(X), d∞ ). Now, the previous proposition
implies that the action G y (C(X), d∞ ) itself has a bounded orbit. Since C(X) is
complete, Lemma 7.2 implies that G has a global fixed point x ∈ C(X). Finally, if
p : C(X) → X denotes the projection given by Theorem 8.10, then p(x) defines a global
fixed point in X. 

8.3

Application to small cancellation quotients

Combining the different results established in the previous sections, we are able to find a
criterion of acylindrical hyperbolicity for groups acting on a small cancellation polygonal
complex. We begin with some preliminaries on acylindrically hyperbolic groups.
34

Definition 8.17. An action by isometries G y (S, d) is acylindrical if, for every d ≥ 0,
there exist two constants R, N ≥ 0 such that, for every x, y ∈ S,
d(x, y) ≥ R ⇒ #{g ∈ G | d(x, gx), d(y, gy) ≤ d} ≤ N .
A group is acylindrically hyperbolic if it admits an action on a (Gromov-)hyperbolic
space which is acylindrical and non-elementary (ie., with an infinite limit set).
In [Osi13], Osin gives several equivalent definitions of the acylindrical hyperbolicity of
a group. The one we will be interested in is (AH3 ) in [Osi13, Theorem 1.2]. This is
because we will use Theorem 7.1 to produce actions on hyperbolic spaces such that any
loxodromic isometries will be WPD.
Definition 8.18. Let G be a group acting by isometries on a metric space (S, d). An
element g ∈ G is WPD if, for every d ≥ 0 and every x ∈ S, there exists M ≥ 1 such
that
#{h ∈ G | d(x, hx), d(g M x, hg M x) ≤ d} < +∞.
Theorem 8.19. [Osi13, Theorem 1.2] A group acting by isometries on a hyperbolic
space with a WPD isometry is either virtually cyclic or acylindrically hyperbolic.
Our main criterion is the following:
Theorem 8.20. Let G be a group acting on a C 0 (1/4) − T (4) polygonal complex X.
Suppose that there exists a constant R ≥ 0 such that, for any points x, y ∈ X, the
intersection stab(x) ∩ stab(y) is finite whenever x and y are separated by at least R
pairwise disjoint hypergraphs. Either the action G y X is elliptic (ie., any element of
G has a fixed a point), or G is virtually cyclic or acylindrically hyperbolic.
The idea is to look at the induced action G y (C(X), d∞ ) on the CAT(0) cube complex
C(X) associated to X. Then, we want to show that the small cancellation properties
of X implies that C(X) does not contain thick grid of hyperplanes, so that (C(X), d∞ )
will be hyperbolic, and then use the assumption on the stabilisers in X to deduce
an acylindrical property for the action on C(X) by applying Theorem 7.1 thanks to
Theorem 8.10. Finally, if we are able to find a loxodromic isometry for the action
G y (C(X), d∞ ), we can deduce that G acts on a hyperbolic space with a WPD
isometry. The fist step of this argument is achieved by the following lemma:
Lemma 8.21. Let X be C 0 (1/4) − T (4) polygonal complex and let C(X) denote its associated CAT(0) cube complex. Then C(X) does not contain (4, 4)-grids of hyperplanes.
Proof. Suppose by contradiction that C(X) contains a (4, 4)-grid of hyperplanes. In
particular, X contains eight hypergraphs V1 , . . . , V4 and H1 , . . . , H4 such that Vi and Hj
are transverse for every 1 ≤ i, j ≤ 4 and Vi (resp. Hi ) separates Vi−1 and Vi+1 (resp.
Hi−1 and Hi+1 ) for i = 2, 3. Let Pi,j be a polygon in which the hypergraphs Vi and Hj
meet. Because P1,1 and P1,4 belong to the hypercarrier Y (V1 ), there exists a sequence
of successively adjacent polygons in Y (V1 ) between P1,1 and P1,4 ; a similar assertion
holds for P1,4 and P4,4 , P4,4 and P4,1 , P4,1 and P1,1 . Let B denote the union of all these
polygons, which we call a band of polygons. A polygon in B will be a corner if it belongs
to {P1,1 , P1,4 , P4,4 , P4,1 } and interior otherwise.
Let γ1 ⊂ B (resp. γ2 , γ3 , γ4 ⊂ B) be a combinatorial path of minimal length between
P1,1 and P1,4 (resp. P1,4 and P4,4 , P4,4 and P4,1 , P4,1 and P1,1 ) strictly included into
the halfcarrier associated to V1 (resp. H4 , V4 , H1 ) which does not contain V4 (resp. H1 ,
V1 , H4 ). Finally, let c1 (resp. c2 , c3 , c4 ) be the combinatorial path of minimal length in
P1,4 (resp. P4,4 , P4,1 , P1,1 ) between γ1 and γ2 (resp. γ2 and γ3 , γ3 and γ4 , γ4 and γ1 ).
We define the loop γ as the concatenation
35

γ = γ1 ∪ c1 ∪ γ2 ∪ c2 ∪ γ3 ∪ c3 ∪ γ4 ∪ c4 .
A vertex of γ will be called angular if it belongs to two polygons of B.

Notice that the intersection between c1 and γ1 or γ2 is a single vertex, because we have
chosen γ1 and γ2 of minimal length; furthermore, c1 is disjoint from c2 , c4 , γ3 and γ4 ,
because they are separated by V2 or H3 . A similar statement holds for c2 , c3 and c4 .
Finally, notice that two γi ’s are always separated by a hypergraph so that they are
pairwise disjoint. We conclude that γ is a simple loop in X. Let D → X be a reduced
non-degenerate disc diagram bounded by γ.
The reciprocal images in D of the hypergraphs V2 , V3 , H2 and H3 must separate D. In
particular, these hypergraphs intersect in different 2-cells in D so that D cannot be a
single 2-cell. According to Theorem 8.3, D necessarily contains a shell; let R denote its
image in X.
We first notice that an interior vertex v of the outer path of R (which is included
into γ) cannot be angular. Otherwise, v belongs to two polygons P1 , P2 ⊂ B and the
triple (P1 , P2 , R) defines a disc diagram with v as an interior vertex of degree three,
contradicting the condition T(4). Therefore, the outer path of R is included into the
boundary of some polygon P ∈ B. If P 6= R, then ∂R is covered by at most three pieces,
contradicting the condition C’(1/4). Thus, R ∈ B.
Case 1: R is a corner. Then R ∈ {P1,1 , P1,4 , P4,4 , P4,1 }, say R = P1,4 , and the outer
path of R is c1 . During the construction of γ, we have chosen c1 so that length(c1 ) ≤
1
1
2 length(∂P1,4 ). Consequently, the length of the inner path of R is at least 2 length(∂R).
By definition, this inner path is covered by at most two pieces, contradicting the condition C’(1/4).
Case 2: R is interior. Then there exist a hypergraph Λ ∈ {V1 , V4 , H1 , H4 } intersecting
two edges of the inner path of R. Because Λ intersects two opposite edges of the polygon
R, this implies that length of the inner path of R is at least 12 length(∂R). By definition,
this inner path is covered by at most two pieces, contradicting the condition C’(1/4).
Therefore, we conclude that C(X) does not contain a (4, 4)-grid of hyperplanes. 
Proof of Theorem 8.20. Let C(X) denote the CAT(0) cube complex associated to X.
According to the previous lemma, C(X) does not contain (4, 4)-grids of hyperplanes,
so we deduce from Theorem 6.1 that (C(X), d∞ ) is hyperbolic. Let x, y ∈ C(X) be
two vertices satisfying d∞ (x, y) ≥ R + 2. If p : C(X) → X denotes the projection
provided by Theorem 8.10, then, because x and y are separated by R + 2 pairwise
disjoint hyperplanes, p(x) and p(y) are separated by R pairwise disjoint hypergraphs,
so that
36

|stab(x) ∩ stab(y)| ≤ |stab(p(x)) ∩ stab(p(y))| < +∞.
We deduce from Theorem 7.1 that any loxodromic isometry of G y (C(X), d∞ ) is WPD.
If any element of G has a bounded orbit in (C(X), d∞ ), then any element has a fixed
a point in C(X), and a fortiori in X using Theorem 8.10: thus, the action G y X is
elliptic. Otherwise, G has an element g with an unbounded orbit in (C(X), d∞ ). In
particular, g is combinatorially hyperbolic, ie., there exists a combinatorial bi-infinite
geodesic γ on which g acts by translations. By assumption, γ has infinite diameter
with respect to d∞ , so, for every n ≥ 1, there exist two vertices x, y ∈ γ which are
separated by at least n pairwise disjoint hyperplanes. Because γ cannot intersect twice
a hyperplane, these hyperplanes separate two subrays of γ; it follows that the Hausdorff
distance between two subrays of γ is not finite. In particular, g is not parabolic with
respect to d∞ : it is a loxodromic isometry.
Therefore, we have proved that G acts on the hyperbolic space (C(X), d∞ ) with a WPD
isometry. We conclude that G is either virtually cyclic or acylindrically hyperbolic. 
The first examples of groups acting on small cancellation polygonal complexes are the
small cancellation groups acting on their Cayley complexes. We develop this example
below.
Lef F be a free group of rank at least two with a fixed free basis S. Every non-trivial
element g of F can be written uniquely as a reduced word g = g1 · · · gn , where each gi
±1
belongs to S ∪ S −1 and gi 6= gi+1
. We refer to the integer n as the length of g, denoted
by |g|. This reduced word is cyclically reduced if g1 6= gn−1 . If two elements h, k ∈ G,
with reduced words h = h1 · · · hr and k = k1 · · · ks , satisfy hr = ks−1 , we say that hn and
ks cancel in the product hk; otherwise, the product is reduced.
Let R ⊂ F be a symmetrised family, ie., R is stable by taking cyclic conjugates
and inverses. Notice that, up to adding all cyclic conjugates of elements of R and their
inverses, we can always suppose that R is symmetrised. An element p ∈ G is a piece
(with respect to R) if there exist two distinct elements r1 , r2 ∈ R such that the products
r1 = pu1 and r2 = pu2 are reduced for some u1 , u2 ∈ G.
Definition 8.22. Let R ⊂ F be a family of cyclically reduced elements and let R denote
its symmetrised. Then R satisfies
• the condition C 0 (λ) if for every piece p and every element r ∈ R, such that the
product r = pu is reduced, we have |p| < λ|r|;
• the condition T (q) if, for every 3 ≤ h < q and every elements r0 , . . . , rh−1 ∈ R
−1
for i ∈ Zh , at least one of the product r1 r2 , . . . , rh−2 rh−1 , rh−1 r1 is
with ri 6= ri+1
reduced.
Theorem 8.23. Infinitely-presented C 0 (1/4)−T (4) groups are acylindrically hyperbolic.
Proof. Let G be an infinitely-presented C 0 (1/4) − T (4) group, ie., G admits a presentation hS | Ri where R is an infinite subset of the free group F (S) satisfying the
condition C 0 (1/4) − T (4). Let X denote the Cayley complex associated to this presentation. Notice that X is a C 0 (1/4) − T (4) polygonal complex, so that G acts freely on a
C 0 (1/4) − T (4) polygonal complex. It follows easily from Theorem 8.20 that G is either
virtually cyclic or acylindrically hyperbolic. On the other hand, as a consequence of
Greendlinger Lemma for C 0 (1/4) − T (4) groups [LS77, Theorem V.4.4], no relation of R
may be deduced from the others, so G is not finitely-presented, and a fortiori G cannot
be virtually cyclic. 
Remark 8.24. Finitely-presented C 0 (1/4) − T (4) are known to be hyperbolic. The
situation is similar for C 0 (1/6) groups: they are hyperbolic in the finitely-presented
case, and acylindrically hyperbolic otherwise [GS14].
37

Example 8.25. In [TV00], the following group presentation is introduced:
GI,k = ha, b | (an bn )k = 1, n ∈ Ii,
with k ≥ 1 and I ⊂ N\{0} infinite. This presentation always satisfies the condition
T (4), and the condition C 0 (1/4) is satisfied precisely when k ≥ 5. Therefore, GI,k is
acylindrically hyperbolic provided that k ≥ 5.
The small cancellation theory we have described has been generalized for graph of groups,
where we are also able to deduce an action on a small cancellation polygonal complex.
Indeed, if G is a graph of groups with R ⊂ π1 (G) a family satisfying some small cancellation condition, then π1 (G) acts on the Bass-Serre tree T associated to G, and, if Ṫ
denotes the usual cone-off of T over the axes of the conjugates of the elements of R,
then the quotient π1 (G)/hhRii acts on the quotient Ṫ /hhRii which is naturally a small
cancellation polygonal complex. Below, we describe the situation for free products.
Lef G = G1 ∗ · · · ∗ Gm be a free product. Every non-trivial element g of G can be
written uniquely as an alternating product g = g1 · · · gn , where each gi is a non-trivial
element of a free factor and no two consecutive gi , gi+1 belong to the same free factor.
The integer n is called the free product length of g, denoted by |g|. This alternating
product is weakly cyclically reduced if |g| ≤ 1 or g1 6= gn−1 . If two elements h, k ∈ G,
with alternating products h = h1 · · · hr and k = k1 · · · ks , satisfy hr = ks−1 , we say that
hn and ks cancel in the product hk; otherwise, the product is weakly reduced.
Let R ⊂ G be a symmetrised family, ie., R is stable by taking cyclic conjugates
and inverses. Notice that, up to adding all cyclic conjugates of elements of R and their
inverses, we can always suppose that R is symmetrised. An element p ∈ G is a piece
(with respect to R) if there exist two distinct elements r1 , r2 ∈ R such that the products
r1 = pu1 and r2 = pu2 are weakly reduced for some u1 , u2 ∈ G.
Definition 8.26. Let R ⊂ G be a family of weakly cyclically reduced elements and let
R denote its symmetrised. Then R satisfies
• the condition C 0 (λ) if for every piece p and every element r ∈ R, such that the
product r = pu is weakly reduced, we have |p| < λ|r|;
• the condition T (4) if the two following conditions are satisfied:
– if r, s, t ∈ R then at least one of the products rs, st, tr is weakly reduced;
– if each of y1 , y2 , y3 is a letter occuring in the alternating products of elements
r, s, t ∈ R, then y1 y2 y3 6= 1 in G.
If R satisfies the condition C 0 (λ), in order to avoid pathological cases, we will use the
convention that |r| ≥ λ for every r ∈ R.
Theorem 8.27. Let G = G1 ∗ · · · ∗ Gn be a finitely-generated free product and R ⊂ G a
family satisfying the condition C 0 (1/4) − T (4). Then the quotient Q = G/hhRii is either
virtually cyclic or acylindrically hyperbolic.
Proof. The group G acts on the Bass-Serre tree T associated to its decomposition as
a free product. Let Ṫ denote the usual cone-off of T over the axes of the conjugates of
the elements of R. Then the quotient Q = G/hhRii naturally acts on X = Ṫ /hhRii.
Naturally, X has the structure of a polygonal complex, and because R satisfies the
condition C 0 (1/4) − T (4), X turns out to be a C 0 (1/4) − T (4) polygonal complex. As a
direct consequence of the well-known Greendlinger Lemma [Sch73, Theorem 1], we get:
Fact 8.28. If w ∈ G\{1} satisfies |w| ≤ 2, then w ∈
/ hhRii.
38

As a consequence, the free factors of G embed in the quotient Q, so that we may identify
them with their images into Q.
Now, we want to apply Theorem 8.20. We first notice that the stabiliser of an edge e in
X is the image of hhRiistab(e0 ) ⊂ G into Q, where the edge e0 ⊂ T is a lift of e in Ṫ ;
because edge-stabilisers in T are trivial, we deduce that edge-stabilisers in X are trivial.
Similarly, if u is a vertex of X and v is a lift in T , then the stabiliser of u in Q is the image
of hhRiistab(v) ⊂ G into Q, ie., it is a conjugate of a free factor. Finally, the stabiliser
of a polygon P in X is the image of hhRiistab(P 0 ) ⊂ G into Q, where the cone P 0 ⊂ Ṫ
is a lift of P in Ṫ . By construction, the basis of the cone P 0 corresponds to the axis γ of
a conjugate f of an element r ∈ R. For every 1 ≤ i ≤ |r| − 1, let gi be an element of the
oriented stabiliser stab+ (γ) such that the translation length of gi is i modulo |r|; if such
an element does not exist, set gi = 1. Now, if g is any element of stab+ (γ) such that its
translation length is k modulo |r|, then the translation length of gk±1 g is a multiple of
|r|, which is also the translation length of f ; because edge-stabilisers in T are trivial, we
conclude that gk±1 g is a power of f , so in particular we have g ∈ gk∓1 hf i. Consequently,
the index of hf i in stab(γ) = stab(P 0 ) is at most 4|r|. A fortiori, since hf i ⊂ hhRii, the
stabiliser of our polygon P in X has cardinality at most 4|r|. We have proved:
Fact 8.29. Edge-stabilisers in X are trivial. The stabiliser of a polygon in X associated
to an element r ∈ R has cardinality at most 4|r|. The vertex-stabilisers in X correspond
to the conjugates of the free factors in Q.
Let x, y ∈ X be two distinct points. We want to prove that the cardinality of stab(x) ∩
stab(y) is bounded above by 4|r| for some r ∈ R. If x or y is not a vertex, then stab(x)
or stab(y) is included into an edge-stabiliser or a polygon-stabiliser, and the conclusion
follows from the previous fact. Therefore, we may suppose that x and y are two vertices.
Let u (resp. v) be a lift of x (resp. y) in T . Notice that, because x 6= y, u and v are
necessarily distinct; in particular, the intersection stab(u) ∩ stab(v) in G is trivial. Now
the intersection stab(x) ∩ stab(y) in Q is precisely the image of
hhRiistab(u) ∩ hhRiistab(v) ⊂ G
in the quotient Q. Let g ∈ G be an element of this intersection. So there exist r1 , r2 ∈
hhRii, g1 ∈ stab(u) and g2 ∈ stab(v) such that r1 g1 = g = r2 g2 . We deduce that
g1 g2−1 = r1−1 r2 ∈ hhRii. It follows from Fact 8.28 that g1 = g2 ∈ stab(u) ∩ stab(v) = {1},
hence g ∈ hhRii. Thus, stab(x) ∩ stab(y) is trivial. We have proved:
Fact 8.30. For any two distinct points x, y ∈ X, there exists some r ∈ R such that
stab(x) ∩ stab(y) has cardinality at most 4|r|.
To conclude, it is sufficient to prove that the action Q y X is not elliptic. Suppose by
contradiction that this is the case. According to Theorem 8.15, Q has a global fixed
point. So Q contains a finite-index subgroup fixing a vertex x ∈ X. Let u be a lift of
x in T . Up to a conjugation, we may suppose that stab(u) is a free factor, say G1 . We
deduce that hhRiiG1 is a finite-index subgroup in G. On the other hand, our previous
fact implies that hhRiiG1 ∩G2 = {1}. Thus, the free product G is finite, a contradiction.

Example 8.31. Let us introduce, for every k, p, r, q, s ≥ 1 and I ⊂ N\{0} infinite, the
following group presentation
HI,k = ha, b, c, d | [(ab)n , (cd)n ]k = 1, ap = bq = cr = ds = 1, n ∈ Ii.
The group HI,k is the quotient of the free product
ha | ap = 1i ∗ hb | bq = 1i ∗ hc | cr = 1i ∗ hd | ds = 1i
39

by the normal closure of the family R = {[(ab)n , (cd)n ]k , n ∈ I}. The condition T (4)
is satisfied if p, q, r, s ≥ 4, and the condition C 0 (1/4) is satisfied if k ≥ 5. Thus, HI,k is
acylindrically hyperbolic provided that p, q, r, s ≥ 4 and k ≥ 5.

8.4

A note on universal actions

Given an acylindrically hyperbolic group G, an element g ∈ G is a generalized loxodromic
element if g is loxodromic with respect to some acylindrical action of G on a hyperbolic
space; equivalently, g ∈ G is a generalized loxodromic element if it is a WPD element
with respect to an action of G on a hyperbolic space [Osi13, Theorem 1.4]. An action of
G on a hyperbolic space S is a universal action if any generalized loxodromic element
of G turns out to be WPD; if G y S is moreover acylindrical, this is an acylindrical
universal action. In [Osi13, Question 6.7], Osin asks whether any finitely-generated
group admits an acylindrical universal action. A negative answer was given in [Abb15],
by proving that Dunwoody’s inaccessible group does not admit an acylindrical universal
action. However, the question is still open for finitely-presented groups.
The action we used for small cancellation quotients satisfies the acylindrical property
given by Theorem 7.1, which implies that any loxodromic isometry turns out to be
WPD. Thus, we deduce:
Theorem 8.32. Any infinitely-presented C 0 (1/4) − T (4) group G admits a universal
action. Furthermore, the generalized loxodromic elements of G are the infinite-order
elements.
Proof. We saw that G acts on the CAT(0) cube complex C(X) dual to the Cayley
complex associated to the small cancellation presentation; furthermore, (C(X), d∞ ) is
hyperbolic, and the action G y (C(X), d∞ ) is free and satisfies the acylindrical property
given by Theorem 7.1. Therefore, we deduce that any infinite-order of G induces a
loxodromic isometry of C(X), and a fortiori a WPD isometry. 
Unfortunately, we do not know if Theorem 7.1 may be improved to deduce a true
acylindricity in general. However such a generalization is possible in finite dimension
(where d∞ may be replaced with the combinatorial distance):
Theorem 8.33. Let G be a group acting on an unbounded hyperbolic CAT(0) cube
complex X. The following are equivalent:
(i) G y X is acylindrical,
(ii) there exist two constants L and R such that, for any vertices x, y ∈ X satisfying
d(x, y) ≥ L, the set {g ∈ G | gx = x, gy = y} has cardinality at most R,
(iii) there exist N and K such that, for any hyperplanes J1 , J2 separated by at least N
hyperplanes, the intersection stab(J1 ) ∩ stab(J2 ) has cardinality at most K.
Proof. Without loss of generality, we may suppose that X is δ-hyperbolic for some
δ ∈ N. According to Theorem 3.3, there exists a constant C such that any grid of
hyperplanes in X is C-thin.
The implication (i) ⇒ (ii) is clear.
Now we prove (ii) ⇒ (iii): we suppose that there exist constants L and R such that,
for any x, y ∈ X satisfying d(x, y) ≥ L, the set {g ∈ G | gx = x, gy = y} has cardinality
at most R. Without loss of generality, we may assume L ≥ max(C + 1, dim(X)). Let
N ≥ Ram(L) and J1 , J2 be two hyperplanes separated by at least N hyperplanes. We
want to prove that |stab(J1 ) ∩ stab(J2 )| is bounded above by a constant which depends
only on R and dim(X).
40

According to Lemma 3.7, there exist L pairwise disjoint hyperplanes V1 , . . . , VL separating J1 and J2 . Let x ∈ N (J1 ) and y ∈ N (J2 ) be two vertices minimizing the distance
between the two neighborhoods N (J1 ) and N (J2 ).
Let H denote the subgroup stab(J1 ) ∩ stab(J2 ) and let g ∈ H. Of course, because
d(x, y) = d(gx, gy), gx and gy minimize also the distance between N (J1 ) and N (J2 ),
so it follows from Proposition 2.7 that exactly the same hyperplanes separate x and y,
and gx and gy, namely those which separate J1 and J2 . If J is a hyperplane separating
x and gx, then it cannot separate x and y or gx and gy, so it has to separate y and
gy, and in particular it has to be transverse to the Vk ’s. Thus, V1 , . . . , VL together
with the hyperplanes separating x and gx define a (L, d∞ (x, gx))-grid of hyperplanes.
Because we have supposed L ≥ C + 1, the definition of C implies d∞ (x, gx) ≤ C, hence
d(x, gx) ≤ C dim(X). Similarly, we prove that d(y, gy) ≤ C dim(X).
We have proved that the orbits of H on N (J1 ) and N (J2 ) are bounded. Because these
neighborhoods are themself CAT(0) cube complexes, this implies that H fixes a point
in each one. Furthermore, H stabilizes the maximal cubes which contain these global
fixed points, so that H contains a subgroup H0 of index at most 2 dim(X)! which fixes
pointwise these two maximal cubes. In particular, H0 has two global fixed vertices a ∈
N (J1 ) and b ∈ N (J2 ). Because V1 , . . . , VL separate J1 and J2 , we deduce that d(a, b) ≥
L. By our hypotheses, this implies that H0 has cardinality at most R. Therefore,
|stab(J1 ) ∩ stab(J2 )| = |H| ≤ 2 dim(X)! · |H0 | ≤ 2R dim(X)!.
This complete the proof.
Now we prove (iii) ⇒ (i): we suppose there exist two constants N and K such that, for
any hyperplanes J1 , J2 separated by at least N hyperplanes, the intersection stab(J1 ) ∩
stab(J2 ) has cardinality at most K. Let  ∈ N\{0} and let x, y ∈ X be two vertices
satisfying
d(x, y) ≥ Ram(N + 2) + 2(C dim(X) +  + Ram(1 + C +  + 8δ)).
We want to prove that the cardinality of the set
F = {g ∈ G | d(x, gx), d(y, gy) ≤ }
is bounded above by a constant which depends only on δ, C, N , K,  and dim(X).
Thus, the acylindricity of the action will follow.
Fix a combinatorial geodesic [x, y] and let z ∈ [x, y] be its midpoint. For convenience,
we will suppose in the following that z is a vertex. Let p, r ∈ [x, z] and q, s ∈ [z, y]
denote the points of [x, y] defined by
d(p, z) = d(q, z) = C dim(X) +  + bRam(N + 2)/2c
and
d(r, z) = d(s, z) = Ram(C +  + 8δ) + C dim(X) +  + bRam(N + 2)/2c.
Along the geodesic [x, y], we fix our points in the following order: x, r, p, z, q, s and
finally y.
Without loss of generality, we may suppose that C +  + 8δ ≥ dim X. Let g ∈ F . We
claim that for all but 2( + C dim(X)) hyperplanes J separating p and q, gJ separates
r and s. If J is a hyperplane separating p and q such that gJ does not separate r and
s, three cases may happen.
Case 1: the hyperplane gJ does not meet [x, y]. Because J separates x and y, gJ must
separate gx and gy. Therefore, since gJ does not separate x and y, necessarily gJ
41

separates either y and gy or x and gx. But d(x, gx), d(y, gy) ≤ , so that there exist at
most 2 such hyperplanes J.
Case 2: the hyperplane gJ meets [s, y]. Let u ∈ [p, q] be a vertex adjacent to J. Then,
because in a δ-hyperbolic space the distance is 8δ-convex and that d(x, gx), d(y, gy) ≤ ,
we deduce that d(u, gu) ≤  + 8δ. Let v ∈ [s, y] be a vertex adjacent to gJ. Because
d(q, s) ≥ Ram(C +  + 8δ), Lemma 3.7 implies that there exist k ≥ 1 + C +  + 8δ
pairwise disjoint hyperplanes V1 , . . . , Vk separating q and s; say that Vi separates Vi−1
and Vi+1 for all 2 ≤ i ≤ k − 1. Then, each hyperplane Vi separates either u and gu or
v and gu; in the latter case, Vi is transverse to gJ. Because d(u, gu) ≤  + 8δ, there are
at most  + 8δ hyperplanes in the first case. On the other hand, because Vi and Vj are
disjoint for any i 6= j, we deduce that, if Vi is transverse to gJ for some i, then Vj is
transverse to gJ for all j ≥ i. Therefore, the hyperplanes V+8δ , . . . , Vk are transverse to
gJ. Thus, because k ≥ C + 1 +  + 8δ, if there exist M hyperplanes J such that gJ meets
[s, y], the images by g of these M hyperlanes (which separate gx and gy) together with
V+8δ , . . . , Vk define a (bM/ dim Xc, C + 1)-grid of hyperplanes. From the definition of
C, we deduce that M ≤ C dim(X).
Case 3: the hyperplane gJ meets [r, x]. This case is symmetric to the previous one.
There are at most C dim(X) such hyperplanes J.
We have just proved that there exist at most 2( + C dim(X)) hyperplanes J separating
p and q such that gJ does not separate r and s, as claimed.
Therefore, if H(a, b) denotes the set of hyperplanes separating two vertices a and b,
any element g ∈ F defines a function (Sg ⊂ H(p, q)) → H(r, s) where H(p, q)\Sg has
cardinality at most 2( + C dim(X)). The cardinality of the set L of these functions is
bounded above by a constant κ which depends only on d(p, q), d(r, s), C and ; from
our choices of d(p, q) and d(r, s), in fact it depends only on δ, C, N ,  and dim(X).
If #F ≥ (K + 1)κ, then F contains K + 1 pairwise distinct elements g1 , . . . , gK+1
inducing the same function of L. Therefore, for all 1 ≤ i ≤ K + 1, g1−1 gi stabilizes
each hyperplane of a family S ⊂ H(p, q) such that H(p, q)\S has cardinality at most
2(+C dim(X)). On the other hand, #H(p, q) = d(p, q) ≥ Ram(N +2)+2(+C dim(X))
hence #S ≥ Ram(N + 2). According to Lemma 3.7, the collection S contains N + 2
pairwise disjoint hyperplanes; because they all separate p and q, it makes sense to
claim the two extremal hyperplanes J1 , J2 of this subcollection are separated by at least
N hyperplanes, hence |stab(J1 ) ∩ stab(J2 )| ≤ K by definition of N . A fortiori, the
T
intersection
stab(J) has cardinality at most K, so we conclude that there exist i 6= j
J∈S

such that g1−1 gi = g1−1 gj , that is gi = gj , a contradiction.
Therefore, #F is bounded above by the constant (K + 1)κ which depends only on δ, C,
N , K,  and dim(X). 
Remark 8.34. The equivalence (i) ⇔ (ii) of Theorem 8.33 has been proved recently in
[Mar15] for hyperbolic CAT(0) square complexes. In his article, Martin introduces the
terminology weakly acylindrical action for an action satisfying the condition (ii), so that
weakly acylindrical actions and acylindrical actions turn out to be equivalent notions in
our context. This statement is especially useful because it is definitely easier to verify
whether an action is weakly acylindrical than acylindrical. Therefore, Theorem 8.33
generalizes [Mar15, Theorem A] to higher dimensions.
Thus, if the CAT(0) cube complex dual to the small cancellation polygonal complex is
finite-dimensional, Theorem 7.1 may be replaced with Theorem 8.33 in the arguments of
the previous section in order to produce an acylindrical action. According to Proposition
8.8, we know that our cube complex is finite-dimensional precisely when the lengths of
42

the elements of the small cancellation family we are considering are uniformly bounded.
Of course, with respect to the classical small cancellation conditions, this may happen
only for finitely-presented groups, which are hyperbolic. On the other hand, for free
products we can state:
Theorem 8.35. Let G be a free product of finitely-generated groups which are neither
acylindrically hyperbolic nor virtually cyclic, and let R ⊂ G be a family satisfying the
condition C 0 (1/4) − T (4) whose elements have uniformly bounded free product lengths.
Then the quotient G/hhRii admits an acylindrical universal action.
Proof. The group G acts on the Bass-Serre tree T associated to its decomposition as
a free product. Let Ṫ denote the usual cone-off of T over the axes of the conjugates of
the elements of R. Then the quotient Q = G/hhRii naturally acts on X = Ṫ /hhRii.
Naturally, X has the structure of a polygonal complex, and because R satisfies the
condition C 0 (1/4) − T (4), X turns out to be a C 0 (1/4) − T (4) polygonal complex. Let
C(X) denote the CAT(0) cube complex associated to X. We know that C(X) does
not contain (4, 4)-grids of hyperplanes and it is finite-dimensional. Thus, according to
Theorem 3.3, C(X) is hyperbolic. Furthermore, combining Fact 8.30 and Theorem 8.33,
we deduce that the action Q y C(X) is acylindrical.
Now, because an element of Q inducing an elliptic isometry of C(X) has to fix a point
in C(X), and so fixes a point in X as a consequence of Theorem 8.10, and so stabilises
a vertex or an edge or a polygon in X, we notice thanks to Fact 8.29 that an element
of Q is elliptic for the action Q y C(X) if and only if it belongs to a conjugate of a
free factor or it has finite order. To conclude our proof, it is sufficient to notice that a
free factor in Q cannot contain a generalized loxodromic element, but this follows from
the assumption that the free factors are neither acylindrically hyperbolic nor virtually
cyclic. 
Example 8.36. For k ≥ 1 and I ⊂ N\{0} infinite, let us introduce the group presentation
Kk,I = ha, b, c, d | [a, b] = [c, d] = 1, (an bn cn dn )k = 1, n ∈ Ii.
The group Kk,I is the quotient of the free product of two free abelian groups
ha, b | [a, b] = 1i ∗ hc, d | [c, d] = 1i
by the normal closure of the family R = {(an bn cn dn )k , n ∈ I}. The condition C 0 (1/4) −
T (4) is satisfied if k ≥ 5. Therefore, Kk,I is acylindrically hyperbolic and admits an
acylindrical universal action whenever k ≥ 5.

References
[Abb15]

Carolyn Abbott. Not all acylindrically hyperbolic groups have universal
acylindrical actions. arXiv:1505.02990, 2015.

[Bah14]

Patrick Bahls. Relative hyperbolicity and right-angled coxeter groups.
arXiv:0401280, 2014.

[BC12]

J. Behrstock and R. Charney. Divergence and quasimorphisms of right-angled
artin groups. Mathematische Annalen, 352(2):339–356, 2012.

[BH99]

Martin R. Bridson and André Haefliger. Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999.
43

[BHSC13] J. Behrstock, M. Hagen, A. Sisto, and P.-E. Caparace. Thickness, relative
hyperbolicity, and randomness in coxeter groups. arXiv:1312.4789, 2013.
[Bow12]

Brian Bowditch. Relatively hyperbolic groups. Internat. J. Algebra and
Computation., 22(3), 2012.

[Bow14]

Brian Bowditch. Uniform hyperbolicity of the curve graphs. Pacific J. Math.,
269:269–280, 2014.

[BvdV91] H.-J. Bandelt and M. van de Vel. Superextensions and the depth of median
graphs. Journal of Combinatorial Theory, Series A, 57:187–202, 1991.
[Cap09]

Pierre-Emmanuel Caprace. Buildings with isolated subspaces and relatively
hyperbolic coxeter groups. Innov. Incidence. Geom., 10:15–31, 2009.

[Che00]

V. Chepoi. Graphs of some CAT(0) complexes.
24(2):125–179, 2000.

[Gen15]

A. Genevois. Hyperbolic diagram groups are free. arXiv:1505.02053, 2015.

[GS14]

D. Gruber and A. Sisto. Infinitely presented graphical small cancellation
groups are acylindrically hyperbolic. arXiv:1408.4488, 2014.

[Hag08]

Frédéric Haglund. Finite index subgroups of graph products. Geom. Dedicata,
135:167–209, 2008.

[Hag14]

Mark F. Hagen. Weak hyperbolicity of cube complexes and quasi-arboreal
groups. J. Topol., 7(2):385–418, 2014.

[HW08]

G. Hruska and D. Wise. Packing subgroups in relatively hyperbolic groups.
Geometry and Topology, 13:1945–1988, 2008.

[Lea13]

Ian J. Leary. A metric kan-thurston theorem. Journal of Topology, 6(1):251–
284, 2013.

[LS77]

C. Lyndon and P. E. Schupp. Combinatorial group theory. Ergebnisse der
Mathematik und ihrer Grenzgebiete 89. Springer, Berlin, 1977.

[Man05]

J. Manning. Geometry of pseudocharacters. Geometry and Topology, 9:1147–
1185, 2005.

[Mar15]

A. Martin. CAT(0) square complexes and acylindricity. arXiv:1509.03131,
2015.

[MW02]

J. McCammond and D. Wise. Fans and ladders in small cancellation theory.
Proc. London Math. Soc., 84(3):599–644, 2002.

[Nib02]

Graham A. Niblo. The singularity obstruction for group splittings. Topology
and its Applications, 119:17–31, 2002.

[Osi13]

D. Osin. Acylindrically hyperbolic groups. arXiv:1304.1246, 2013.

[Pap95]

P. Papasoglu. Strongly geodesically automatic groups are hyperbolic. Inventiones Mathematicae, 121(2):323–334, 1995.

[Pol93]

N. Polat. Finite invariant simplices in infinite graphs. Periodica Mathematica
Hungarica, 27(2):125–136, 1993.

[Sag95]

Michah Sageev. Ends of group pairs and non-positively curved cube complexes. Proc. London Math. Soc. (3), 71(3):585–617, 1995.
44

Adv. in Appl. Math.,

[Sch73]

P. E. Schupp. A survey of small cancellation. In Word Problems (Amsterdam:
Noth-Holland), pages 569–589. 1973.

[Sig13]

José M. Sigarreta. Hyperbolicity in median graphs. Proc. Indian Acad. Sci.
Math. Sci., 123(4):455–467, 2013.

[TV00]

S. Thomas and B. Velickovic. Asymptotic cones of finitely generated groups.
Bull. London Math. Soc., 32(2):203–208, 2000.

[VK33]

Egbert Van Kampen. On some lemmas in the theory of groups. American
Journal of Mathematics, 55(1):268–273, 1933.

[Wis04]

D. T. Wise. Cubulating small cancellation groups. Geom. Funct. Anal.,
14(1):150–214, 2004.

[Wis12]

Daniel T. Wise.
preprint, 2012.

The structure of groups with a quasiconvex hierarchy.

45

