ON HIERARCHICAL HYPERBOLICITY OF CUBICAL GROUPS

arXiv:1609.01313v2 [] 23 Jan 2018

MARK F HAGEN AND TIM SUSSE

Abstract. Let X be a proper CAT(0) cube complex admitting a proper cocompact action
by a group G. We give three conditions on the action, any one of which ensures that X has a
factor system in the sense of [BHS14]. We also prove that one of these conditions is necessary.
This combines with [BHS14] to show that G is a hierarchically hyperbolic group; this partially
answers questions raised in [BHS14, BHS15]. Under any of these conditions, our results also
affirm a conjecture of Behrstock-Hagen on boundaries of cube complexes, which implies that
X cannot contain a convex staircase. The necessary conditions on the action are all strictly
weaker than virtual cospecialness, and we are not aware of a cocompactly cubulated group that
does not satisfy at least one of the conditions.

Introduction
Much work in geometric group theory revolves around generalizations of Gromov hyperbolicity: relatively hyperbolic groups, weakly hyperbolic groups, acylindrically hyperbolic groups,
coarse median spaces, semihyperbolicity, lacunary hyperbolicity, etc. Much attention has been
paid to groups acting properly and cocompactly on CAT(0) cube complexes, which also have
features reminiscent of hyperbolicity. Such complexes give a combinatorially and geometrically
rich framework to build on, and many groups have been shown to admit such actions (for a
small sample, see [Sag95, Wis04, OW11, BW12, HW15]).
Many results follow from studying the geometry of CAT(0) cube complexes, often using strong
properties reminiscent of negative curvature. For instance, several authors have studied the
structure of quasiflats and Euclidean sectors in cube complexes, with applications to rigidity
properties of right-angled Artin group [Xie05, BKS08, Hua14]. These spaces have also been
shown to be median [Che00] and to have only semi-simple isometries [Hag07]. Further, under
reasonable assumptions, a CAT(0) cube complex X either splits as a nontrivial product or
Isom(X ) must contain a rank-one element [CS11]. Once a given group is known to act properly
and cocompactly on a CAT(0) cube complex the geometry of the cube complex controls the
geometry and algebra of the group. For instance, such a group is biautomatic and cannot have
Kazhdan’s property (T) [NR98, NR97], and it must satisfy a Tits alternative [SW05]
Here, we examine cube complexes admitting proper, cocompact group actions from the point
of view of certain convex subcomplexes. Specifically, given a CAT(0) cube complex X , we study
the following set F of convex subcomplexes: F is the smallest set of subcomplexes that contains
X , contains each combinatorial hyperplane, and is closed under cubical closest-point projection,
i.e. if A, B ∈ F, then gB (A) ∈ F, where gB : X → B is the cubical closest point projection.
Main results. The collection F of subcomplexes is of interest for several reasons. It was
first considered in [BHS14], in the context of finding hierarchically hyperbolic structures on X .
Specifically, in [BHS14], it is shown that if there exists N < ∞ so that each point of X is
contained in at most N elements of F, then X is a hierarchically hyperbolic space, which has
Date: January 24, 2018.
Hagen was supported by the Engineering and Physical Sciences Research Council grant of Henry Wilton.
Susse was partially supported by National Science Foundation grant DMS-1313559.
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ON HIERARCHICAL HYPERBOLICITY OF CUBICAL GROUPS

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numerous useful consequences outlined below; the same finite multiplicity property of F has
other useful consequences outlined below. When this finite multiplicity condition holds, we say,
following [BHS14], that F is a factor system for X .
We believe that if X is proper and some group G acts properly and cocompactly by isometries
on X , then the above finite multiplicity property holds, and thus G is a hierarchically hyperbolic
group. In [BHS14], it is shown that this holds when G has a finite-index subgroup acting
cospecially on X , and it is also verified in a few non-cospecial examples.
This conjecture has proved surprisingly resistant to attack; we earlier believed we had a proof.
However, a subtlety in Proposition 5.1 means that at present our techniques only give a complete
proof under various conditions on the G–action, namely:
Theorem A. Let G act properly and cocompactly on the proper CAT(0) cube complex X . Then
F is a factor system for X provided any one of the following conditions is satisfied (up to
passing to a finite-index subgroup of G):
• the action of G on X is rotational;
• the action of G on X satisfies the weak height condition for hyperplanes;
• the action of G on X satisfies the essential index condition and the Noetherian intersection of conjugates condition (NICC) on hyperplane-stabilisers.
Hence, under any of the above conditions, X is a hierarchically hyperbolic space and G a hierarchically hyperbolic group.
Conversely, if F is a factor system, then the G–action satisfies the essential index condition
and the NICC.
The auxiliary conditions are as follows. The action of G is rotational if, whenever A, B are
hyperplanes of X , and g ∈ StabG (B) has the property that A and gA cross or osculate, then A
lies at distance at most 1 from B. This condition is prima facie weaker than requiring that the
action of G on X be cospecial, so Theorem A generalises the results in [BHS14]. (In fact, the
condition above is slightly stronger than needed; compare Definition 4.1.)
A subgroup K ≤ G satisfies the weak T
finite height condition if the following holds. Let
{gi }i∈I ⊂ G be an infinite set so that K ∩ i∈J K gi is infinite for all finite J ⊂ I. Then there
exist distinct gi , gj so that K ∩ K gi = K ∩ K gj . The action of G on X satisfies the weak height
condition for hyperplanes if each hyperplane stabiliser satisfies the weak height condition.
This holds, for example, when each hyperplane stabiliser has finite height in the sense
of [GMRS98]. Hence Theorem A implies that F is a factor system when X is hyperbolic,
without invoking virtual specialness [Ago13] because quasiconvex subgroups (in particular hyperplane stabilisers) have finite height [GMRS98]; the existence of a hierarchically hyperbolic
structure relative to F also follows from recent results of Spriano in the hyperbolic case [Spr17].
Also, if F is a factor system and X does not decompose as a product of unbounded CAT(0) cube
complexes, then results of [BHS14] imply that G is acylindrically hyperbolic. On the other hand,
recent work of Genevois [Gen16] uses finite height of hyperplane-stabilisers to verify acylindrical
hyperbolicity for certain groups acting on CAT(0) cube complexes. In our opinion, this provides
some justification for the naturality of the weak height condition for hyperplanes.
The NIC condition for hyperplanes
asks the following for each hyperplane-stabiliser K. Given
T
any {gi }i≥0 so that Kn = K ∩ ni=0 K gi is infinite for all n, there exists ` so that Kn and K`
are commensurable for n ≥ `. Note that ` is allowed to depend on {gi }i≥0 . The accompanying
essential index condition asks that there exists a constant ζ so that for any F ∈ F , the stabiliser
of F has index at most ζ in the stabiliser of the essential core of F , defined in [CS11]. These
conditions are somewhat less natural than the preceding conditions, but they follow fairly easily
from the finite multiplicity of F.
We prove Theorem A in Section 6. There is a unified argument under the weak finite height
and NICC hypotheses, and a somewhat simpler argument in the presence of a rotational action.

ON HIERARCHICAL HYPERBOLICITY OF CUBICAL GROUPS

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To prove Theorem A, the main issue is to derive a contradiction from the existence of an
infinite strictly ascending chain {Fi }, in F, using that the corresponding chain of orthogonal
complements must strictly descend. The existence of such chains can be deduced from the
failure of the finite multiplicity of F using only the proper cocompact group action; it is in
deriving a contradiction from the existence of such chains that the other conditions arise.
Any condition that allows one to conclude that the Fi have bounded-diameter fundamental
domains for the actions of their stabilisers yields the desired conclusion. So, there are most
likely other versions of Theorem A using different auxiliary hypotheses. We are not aware of a
cocompactly cubulated group which is not covered by Theorem A.
Hierarchical hyperbolicity. Hierarchically hyperbolic spaces/groups (HHS/G’s), introduced
in [BHS14, BHS15], were proposed as a common framework for studying mapping class groups
and (certain) cubical groups. Knowledge that a group is hierarchically hyperbolic has strong
consequences that imply many of the nice properties of mapping class groups.
Theorem A and results of [BHS14] (see Remark 13.2 of that paper) together answer Question
8.13 of [BHS14] and part of Question A of [BHS15] — which ask whether a proper cocompact
CAT(0) cube complex has a factor system — under any of the three auxiliary hypotheses
in Theorem A. Hence our results expand the class of cubical groups that are known to be
hierarchically hyperbolic. Some consequences of this are as follows, where X is a CAT(0) cube
complex on which G acts geometrically, satisfying any of the hypotheses in Theorem A:
• In combination with [BHS14, Corollary 14.5], Theorem A shows that G acts acylindrically on the contact graph of X , i.e. the intersection graph of the hyperplane carriers,
which is a quasi-tree [Hag14].
• Theorem A combines with Theorem 9.1 of [BHS14] to provide a Masur-Minsky style
distance estimate in G: up to quasi-isometry, the distance in X from x to gx, where
g ∈ G, is given by summing the distances between the projections of x, gx to a collection
of uniform quasi-trees associated to the elements of the factor system.
• Theorem A combines with Corollary 9.24 of [DHS16] to prove that either G stabilizes
a convex subcomplex of X splitting as the product of unbounded subcomplexes, or G
contains an element acting loxodromically on the contact graph of X . This is a new
proof of a special case of the Caprace-Sageev rank-rigidity theorem [CS11].
Proposition 11.4 of [BHS14] combines with Theorem A to prove:
Theorem B. Let G act properly and cocompactly on the proper CAT(0) cube complex X , with
the action satisfying the hypotheses of Theorem A. Let F be the factor system, and suppose that
for all subcomplexes A ∈ F and g ∈ G, the subcomplex gA is not parallel to a subcomplex in F
which is in the orthogonal complement of A. Then X quasi-isometrically embeds in the product
of finitely many trees.
The set F is shown in Section 2 to have a graded structure: the lowest-grade elements are
combinatorial hyperplanes, then we add projections of combinatorial hyperplanes to combinatorial hyperplanes, etc. This allows for several arguments to proceed by induction on the grade.
Essentially by definition, a combinatorial hyperplane H is the orthogonal complement of a 1–
cube e, i.e. a maximal convex subcomplex H for which X contains the product e × H as a
subcomplex. We show, in Theorem 3.3, that F is precisely the set of convex subcomplexes F
such that there exists a compact, convex subcomplex C so that the orthogonal complement of
C is F . This observation plays an important role.
Relatedly, we give conditions in Proposition 5.1 ensuring that F is closed under the operation of taking orthogonal complements. As well as being used in the proof of Theorem A,
this is needed for applications of recent results about hierarchically hyperbolic spaces to cube

ON HIERARCHICAL HYPERBOLICITY OF CUBICAL GROUPS

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complexes. Specifically, in [ABD17], Abbott-Behrstock-Durham introduce hierarchically hyperbolic spaces with clean containers, and work under that (quite natural) hypothesis. Among
its applications, they produce largest, universal acylindrical actions on hyperbolic spaces for
hierarchically hyperbolic groups. We will not give the definition of clean containers for general
hierarchically hyperbolic structures, but for the CAT(0) cubical case, our results imply that it
holds for hierarchically hyperbolic structures on X obtained using F, as follows:
Theorem C (Clean containers). Let X be a proper CAT(0) cube complex on which the group
G acts properly and cocompactly, and suppose F is a factor system. Let F ∈ F, and let V ∈ F
be properly contained in F . Then there exists U ∈ F, unique up to parallelism, such that:
• U ⊂ F;
• V ,→ F extends to a convex embedding V × U ,→ F ;
• if W ∈ F, and the above two conditions hold with U replaced by W , then W is parallel
to a subcomplex of U .
Proof. Let x ∈ V be a 0–cube and let U 0 = V ⊥ , the orthogonal complement of V at x (see
Definition 1.10). Proposition 5.1 implies that U 0 ∈ F, so U = U 0 ∩ F is also in F, since F is
closed under projections. By the definition of the orthogonal complement, V → X extends to
a convex embedding V × U 0 → X , and (V × U 0 ) ∩ F = V × U since V ⊂ F and F, V × U 0 are
convex. Now, if W ∈ F and W ⊂ F , and V → X extends to a convex embedding V × W → X ,
then V × W is necessarily contained in F , by convexity. On the other hand, by the definition
of the orthogonal complement, W is parallel to a subcomplex of U 0 . Hence W is parallel to a
subcomplex of U . This implies the third assertion and uniqueness of U up to parallelism.

We now turn to applications of Theorem A that do not involve hierarchical hyperbolicity.
Simplicial boundary and staircases. Theorem A also gives insight into the structure of the
boundary of X . We first mention an aggravating geometric/combinatorial question about cube
complexes which is partly answered by our results.
A staircase is a CAT(0) cube complex Z defined as follows. First, a ray-strip is a square
complex of the form Sn = [n, ∞) × [− 12 , 12 ], with the product cell-structure where [n, ∞) has
0–skeleton {m ∈ Z : m ≥ n} and [− 12 , 12 ] is a 1–cube. To build Z, choose an increasing sequence
(an )n of integers, collect the ray-strips San ∼
= [an , ∞) × [− 21 , 12 ], and identify [an+1 , ∞) × {− 12 } ⊂
1
San +1 with [an , ∞) × { 2 } ⊂ San for each n. The model staircase is the cubical neighbourhood
of a Euclidean sector in the standard tiling of E2 by squares, with one bounding ray in the
x–axis, but for certain (an )n , Z may not contain a nontrivial Euclidean sector. (One can define
a d-dimensional staircase analogously for d ≥ 2.) We will see below that the set of “horizontal”
hyperplanes in Z – see Figure 1 for the meaning of “horizontal” – is interesting because there is
no geodesic ray in Z crossing exactly the set of horizontal hyperplanes.

Figure 1. Part of a staircase.
Now let X be a proper CAT(0) cube complex with a group G acting properly and cocompactly.
Can there be a convex staircase subcomplex in X ? A positive answer seems very implausible,
but this question is open and has bothered numerous researchers.
In Section 7, we prove that if F is a factor system, then X cannot contain a convex staircase.
Hence, if X admits a geometric group action satisfying any of the hypotheses in Theorem A,

ON HIERARCHICAL HYPERBOLICITY OF CUBICAL GROUPS

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then X cannot contain a convex staircase. In fact, we prove something more general, which is
best formulated in terms of the simplicial boundary ∂4 X .
Specifically, the simplicial boundary ∂4 X of a CAT(0) cube complex X was defined in [Hag13].
Simplices of ∂4 X come from equivalence classes of infinite sets H of hyperplanes such that:
• if H, H 0 ∈ H are separated by a hyperplane V , then V ∈ H;
• if H1 , H2 , H3 ∈ H are disjoint, then one of H1 , H2 , H3 separates the other two;
• for H ∈ H, at most one halfspace associated to H contains infinitely many V ∈ H.
These boundary sets are partially ordered by coarse inclusion (i.e., A  B if all but finitely
many hyperplanes of A are contained in B), and two are equivalent if they have finite symmetric
difference; ∂4 X is the simplicial realization of this partial order. The motivating example of
a simplex of ∂4 X is: given a geodesic ray γ of X , the set of hyperplanes crossing γ has the
preceding properties. Not all simplices are realized by a geodesic ray in this way: a simplex in
X is called visible if it is. For example, if Z is a staircase, then ∂4 Z has an invisible 0–simplex,
represented by the set of horizontal hyperplanes.
Conjecture 2.8 of [BH16] holds that every simplex of ∂4 X is visible when X admits a proper
cocompact group action; Theorem A hence proves a special case. Slightly more generally:
Corollary D. Let X be a proper CAT(0) cube complex which admits a proper and cocompact
group action satisfying the NICC for hyperplanes. Then every simplex is ∂4 X is visible. Moreover, let v ∈ ∂4 X be a 0–simplex. Then there exists a CAT(0) geodesic ray γ such that the set
of hyperplanes crossing γ represents v.
The above could, in principle, hold even if F is not a factor system, since we have not imposed
the essential index condition. The “moreover” part follows from the first part and [Hag13,
Lemma 3.32]. Corollary D combines with [BH16, Theorem 5.13] to imply that ∂4 X detects
thickness of order 1 and quadratic divergence for G, under the NICC condition. Corollary D
also implies the corollary about staircases at the beginning of this paper. More generally, we
obtain the following from Corollary D and a simple argument in [Hag13]:
Corollary E. Let γ be a CAT(0)-metric or combinatorial geodesic ray in X , where X is as in
Corollary D and the set of hyperplanes crossing γ represents a d-dimensional simplex of ∂4 X .
Then there exists a combinatorially isometrically embedded d+1-dimensional orthant subcomplex
O ⊆ Hull(γ). Moreover, γ lies in a finite neighbourhood of O.
(A k-dimensional orthant subcomplex is a CAT(0) cube complex isomorphic to the product of
k copies of the standard tiling of [0, ∞) by 1–cubes, and the convex hull Hull(A) of a subspace
A ⊆ X is the smallest convex subcomplex containing A.)
Corollary E is related to Lemma 4.9 of [Hua14] and to statements in [Xie05, BKS08] about
Euclidean sectors in cocompact CAT(0) cube complexes and arcs in the Tits boundary. In
particular it shows that in any CAT(0) cube complex with a proper cocompact group action
satisfying NICC, nontrivial geodesic arcs on the Tits boundary extend to arcs of length π/2.
Further questions and approaches. We believe that any proper cocompact CAT(0) cube
complex admits a factor system, but that some additional ingredient is needed to remove the
auxiliary hypotheses in Theorem A; we hope that the applications we have outlined stimulate
interest in finding this additional idea. Since the property of admitting a factor system is
inherited by convex subcomplexes [BHS14], we suggest trying to use G–cocompactness of X to
arrange for a convex (non-G–equivariant) embedding of X into a CAT(0) cube complex Y where
a factor system can be more easily shown to exist. One slightly outrageous possibility is:
Question 1. Let X be a CAT(0) cube complex which admits a proper and cocompact group
action. Does X embed as a convex subcomplex of the universal cover of the Salvetti complex
of some right-angled Artin group?

ON HIERARCHICAL HYPERBOLICITY OF CUBICAL GROUPS

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However, there are other possibilities, for example trying to embed X convexly in a CAT(0)
cube complex whose automorphism group is sufficiently tame to enable one to use the proof of
Theorem A, or some variant of it.
Plan of the paper. In Section 1, we discuss the necessary background. Section 2 contains
basic facts about F, and Section 3 relates F to orthogonal complements. Section 4 introduces
the auxiliary hypotheses for Theorem A, which we prove in Section 6. The applications to the
simplicial boundary are discussed in Section 7.
Acknowledgements. MFH thanks: Jason Behrstock and Alessandro Sisto for discussions of
factor systems during our work on [BHS14]; Nir Lazarovich and Dani Wise for discussions on
Question 1; Talia Fernós, Dan Guralnik, Alessandra Iozzi, Yulan Qing, and Michah Sageev for
discussions about staircases. We thank Richard Webb and Henry Wilton for helping to organize
Beyond Hyperbolicity (Cambridge, June 2016), at which much of the work on this paper was
done. Both of the authors thank Franklin for his sleepy vigilance and Nir Lazarovich for a
comment on the proof of Lemma 2.9. We are greatly indebted to Bruno Robbio and Federico
Berlai for a discussion which led us to discover a gap in an earlier version of this paper, and to
Elia Fioravanti for independently drawing our attention to the same issue. We also thank an
anonymous referee for helpful comments on an earlier version.
1. Background
1.1. Basics on CAT(0) cube complexes. Recall that a CAT(0) cube complex X is a simplyconnected cube complex in which the link of every vertex is a simplicial flag complex (see
e.g. [BH99, Chapter II.5], [Sag14, Wis, Che00] for precise definitions and background). In this
paper, X always denotes a CAT(0) cube complex. Our choices of language and notation for
describing convexity, hyperplanes, gates, etc. follow the account given in [BHS14, Section 2].
Definition 1.1 (Hyperplane, carrier, combinatorial hyperplane). A midcube in the unit cube
c = [− 12 , 12 ]n is a subspace obtained by restricting exactly one coordinate to 0. A hyperplane in
X is a connected subspace H with the property that, for all cubes c of X , either H ∩ c = ∅ or
H ∩c consists of a single midcube of c. The carrier N (H) of the hyperplane H is the union of all
closed cubes c of X with H ∩c 6= ∅. The inclusion H → X extends to a combinatorial embedding
∼
=
H × [− 21 , 21 ] −→ N (H) ,→ X identifying H × {0} with H. Now, H is isomorphic to a CAT(0)
cube complex whose cubes are the midcubes of the cubes in N (H). The subcomplexes H ± of
N (H) which are the images of H ×{± 12 } under the above map are isomorphic as cube complexes
to H, and are combinatorial hyperplanes in X . Thus each hyperplane of X is associated to two
combinatorial hyperplanes lying in N (H).
Remark. The distinction between hyperplanes (which are not subcomplexes) and combinatorial hyperplanes (which are) is important. Given A ⊂ X , either a convex subcomplex or a
hyperplane, and a hyperplane H, we sometimes say H crosses A to mean that H ∩ A 6= ∅.
Observe that the set of hyperplanes crossing a hyperplane H is precisely the set of hyperplanes
crossing the associated combinatorial hyperplanes.
Definition 1.2 (Convex subcomplex). A subcomplex Y ⊆ X is convex if Y is full — i.e. every
cube c of X whose 0–skeleton lies in Y satisfies c ⊆ Y — and Y (1) , endowed with the obvious
path-metric, is metrically convex in X (1) .
There are various characterizations of cubical convexity. Cubical convexity coincides with
CAT(0)–metric convexity for subcomplexes [Hag07], but not for arbitrary subspaces.
Definition 1.3 (Convex Hull). Given a subset A ⊂ X , we denote by Hull(A) its convex hull,
i.e. the intersection of all convex subcomplexes containing A.

ON HIERARCHICAL HYPERBOLICITY OF CUBICAL GROUPS

7

If Y ⊆ X is a convex subcomplex, then Y is a CAT(0) cube complex whose hyperplanes have
the form H ∩ Y, where H is a hyperplane of X , and two hyperplanes H ∩ Y, H 0 ∩ Y intersect if
and only if H, H 0 intersect.
Recall from [Che00] that the graph X (1) , endowed with the obvious path metric dX in which
edges have length 1, is a median graph (and in fact being a median graph characterizes 1–skeleta
of CAT(0) cube complexes among graphs): given 0–cubes x, y, z, there exists a unique 0–cube
m = m(x, y, z), called the median of x, y, z, so that Hull(x, y) ∩ Hull(y, z) ∩ Hull(x, z) = {m}.
Let Y ⊆ X be a convex subcomplex. Given a 0–cube x ∈ X , there is a unique 0–cube y ∈ Y
so that dX (x, y) is minimal among all 0–cubes in Y. Indeed, if y 0 ∈ Y, then the median m of
x, y, y 0 lies in Y, by convexity of Y, but dX (x, y 0 ) = dX (x, m) + dX (m, y 0 ), and the same is true
for y. Thus, if dX (x, y 0 ) and dX (x, y) realize the distance from x to Y (0) , we have m = y = y 0 .
Definition 1.4 (Gate map on 0–skeleton). For a convex subcomplex Y ⊆ X , the gate map to
Y is the map gY : X (0) → Y (0) so that or all v ∈ X (0) , gY (v) is the unique 0–cube of Y lying
closest to v.
Lemma 1.5. Let Y ⊆ X be a convex subcomplex. Then the map gY from Definition 1.4 extends
to a unique cubical map gY : X → Y so that the following holds: for any d–cube c, of X with
vertices x0 , . . . , x2d ∈ X (0) , the map gY collapses c to the unique k–cube c0 in Y with 0–cells
gY (x0 ) . . . , gY (x2d ) in the natural way, respecting the cubical structure.
Furthermore, for any convex subcomplex Y, Y 0 ⊆ X , the hyperplanes crossing gY (Y 0 ) are
precisely the hyperplanes which cross both Y and Y 0 .
Proof. The first part is proved in [BHS14, p. 1743]: observe that the integer k is the number
of hyperplanes that intersect both c and Y. The hyperplanes that intersect c0 are precisely the
hyperplanes which intersect both c and Y. Indeed, the Helly property ensures that there are
cubes crossing exactly this set of hyperplanes, while convexity of Y shows that at least one such
cube lies in Y; the requirement that it contain gY (xi ) then uniquely determines c0 .
To prove the second statement, let H be a hyperplane crossing Y and Y 0 . Then H separates
0–cubes y1 , y2 ∈ Y 0 , and thus separates their gates in Y, since, because it crosses Y, it cannot
separate y1 or y2 from Y. On the other hand, if H crosses gY (Y 0 ), then it separates gY (y1 ), gY (y2 )
for some y1 , y2 ∈ Y 0 . Since it cannot separate y1 or y2 from Y, the hyperplane H must separate
y1 from y2 and thus cross Y. (Here we have used the standard fact that H separates yi from
gY (yi ) if and only if H separates yi from Y; see e.g. [BHS14, p. 1743].) Hence H crosses gY (Y 0 )
if and only if H crosses Y, Y 0 .

The next definition formalizes the relationship between gY (Y 0 ), gY 0 (Y) in the above lemma.
Definition 1.6 (Parallel). The convex subcomplexes F and F 0 are parallel, written F k F 0 , if
for each hyperplane H of X , we have H ∩ F 6= ∅ if and only if H ∩ F 0 6= ∅. The subcomplex
F is parallel into F 0 if F is parallel to a subcomplex of F 0 , i.e. every hyperplane intersecting F
intersects F 0 . We denote this by F ,→k F 0 . Any two 0–cubes are parallel subcomplexes.
The following is proved in [BHS14, Section 2] and illustrated in Figure 2:
Lemma 1.7. Let F, F 0 be parallel subcomplexes of the CAT(0) cube complex X . Then Hull(F ∪
F 0) ∼
= F × A, where A is the convex hull of a shortest combinatorial geodesic with endpoints on
F and F 0 . The hyperplanes intersecting A are those separating F, F 0 . Moreover, if D, E ⊂ X
are convex subcomplexes, then gE (D) ⊂ E is parallel to gD (E) ⊂ D.
The next Lemma will be useful in Section 2.
Lemma 1.8. For convex subcomplexes C, D, E, we have ggC (D) (E) k gC (gD (E)) k gC (gE (D)).

ON HIERARCHICAL HYPERBOLICITY OF CUBICAL GROUPS

gD (E)

8

gE (D)

H

D

V

E

Figure 2. Here, D, E are convex subcomplexes. The gates gD (E), gE (D) are
parallel, and are joined by a product region, shown as a cylinder. Each hyperplane crossing Hull gD (E) ∪ gE (D) either separates gD (E), gE (D) (e.g. the
hyperplane V ) or crosses both of gD (E), gE (D) (e.g. the hyperplane H).
Proof. Let F = gC (D). Let H be a hyperplane so that H ∩ gF (E) 6= ∅. Then H ∩ E, H ∩ F 6= ∅
and thus H ∩ C, H ∩ D 6= ∅, by Lemma 1.5. Thus gF (E) is parallel into gC (gD (E)) and
gC (gE (D)). However, the hyperplanes crossing either of these are precisely the hyperplanes
crossing all of C, D, E. Thus, they cross F and D, and thus cross gF (E) by Lemma 1.5.

The next lemma will be used heavily in Section 6, and gives a group theoretic description of
the stabilizer of a projection.
Lemma 1.9. Let X be a proper CAT(0) cube complex on which G acts properly and cocompactly.
Let H, H 0 be two convex subcomplexes in X . Then StabG (gH (H 0 )) is commensurable with
StabG (H) ∩ StabG (H 0 ). Further, for any finite collection H1 , . . . , Hn of convex subcomplexes
whose
stabilisers act cocompactly, StabG (gH1 (gH2 (· · · gHn−1 (Hn ) · · · ))) is commensurable with
Tn
Stab
G (Hi ).
i=1
Proof. Let H and H 0 be two convex subcomplexes and suppose that g ∈ StabG (H) ∩ StabG (H 0 ).
Then g ∈ StabG (gH (H 0 )), and thus StabG (H) ∩ StabG (H 0 ) ≤ StabG (gH (H 0 )).
Let d = d(H, H 0 ). In particular, for any 0–cube in gH (H 0 ), its distance to H 0 is exactly
d. However, there are only finitely many such translates of H 0 in X , and any element of
StabG (gH (H 0 )) must permute these. Further, there are only finitely many translates of H in X
that contain gH (H 0 ), and any element of the stabilizer must also permute those. Thus, there
is a finite index subgroup (obtained as the kernel of the permutation action on the finite sets
of hyperplanes) that stabilizes both H and H 0 . A similar argument covers the case of finitely
many complexes.

Definition 1.10 (Orthogonal complement). Let A ⊆ X be a convex subcomplex. Let PA be the
convex hull of the union of all parallel copies of A, so that PA ∼
= A × A⊥ , where A⊥ is a CAT(0)
cube complex that we call the abstract orthogonal complement of A in X . Let φA : A × A⊥ → X
be the cubical isometric embedding with image PA .
For any a ∈ A(0) , the convex subcomplex φA ({a} × A⊥ ) is the orthogonal complement of A
at a. See Figures 3 and 4.
Lemma 1.11. Let A ⊆ X be a convex subcomplex. For any a ∈ A, a hyperplane H intersects
φA ({a} × A⊥ ) if and only if H is disjoint from every parallel copy of A but intersects each
hyperplane V with V ∩ A 6= ∅. Hence φA ({a} × A⊥ ), φA ({b} × A⊥ ) are parallel for all a, b ∈ A(0) .
Proof. This follows from the definition of PA : the hyperplanes crossing PA are partitioned into
two classes, those intersecting A (and its parallel copies) and those disjoint from A (and any of
its parallel copies). By definition, φA ({a} × A⊥ ) is the convex hull of the set of 0–cubes of PA
that are separated from a only by hyperplanes of the latter type. The product structure ensures
that any hyperplane of the first type crosses every hyperplane of the second type.


ON HIERARCHICAL HYPERBOLICITY OF CUBICAL GROUPS

9

e⊥

e
Figure 3. Combinatorial hyperplanes are orthogonal complements of 1–cubes.
e⊥
1

s⊥
e⊥
2

⊥

(e1 ∪ s ∪ e2 )

e1

e2
s

Figure 4. Orthogonal complements of 1–cubes e1 , e2 and 2–cube s are shown.
Note that (e1 ∪ e2 ∪ s)⊥ k ge⊥ (ge⊥ (s⊥ )).
2

1

Finally, in [CS11], Caprace and Sageev defined the notion of an essential hyperplane and an
essential action. We record the necessary facts here.
Definition 1.12. Let X be a CAT(0) cube complex, and let F ⊆ X be a convex subcomplex.
Let G ≤ Aut(X ) preserve F .
(1) We say that a hyperplane H is essential in F if H crosses F , and each halfspace associated to H contains 0–cubes of F which are arbitrarily far from H.
(2) We say that H if G–essential in F if for any 0–cube x ∈ F , each halfspace associated
to H contains elements of Gx arbitrarily far from H.
(3) We say that G acts essentially on F if every hyperplane crossing F is G–essential in F .
If G acts cocompactly on F , then a hyperplane is G–essential if and only if it is essential in F .
Proposition 1.13. Let X be a proper CAT(0) cube complex admitting a proper cocompact action
by a group Γ, let F ⊆ X be a convex subcomplex, and let G ≤ Γ act on F cocompactly. Then:
(i) there exists a G–invariant convex subcomplex FbG , called the G–essential core of F ,
crossed by every essential hyperplane in F , on which G acts essentially and cocompactly;
(ii) FbG is unbounded if and only if F is unbounded;
(iii) if G0 ≤ Aut(X ) also acts on F cocompactly, then FbG0 is parallel to FbG ;
(iv) if G0 ≤ G is a finite-index subgroup, we can take FbG0 = FbG .
(v) the subcomplex FbG is finite Hausdorff distance from F .
Proof. By [CS11, Proposition 3.5], F contains a G–invariant convex subcomplex FbG on which
G acts essentially and cocompactly (in particular, FbG is unbounded if and only if F is, and
dHaus (F, FbG ) < ∞). The hyperplanes of F crossing FbG are precisely the G–essential hyperplanes. Observe that if H is a hyperplane crossing F essentially, then cocompactness of the
G–action on F implies that H is G–essential and thus crosses FbG . It follows that if G, G0 both
act on F cocompactly, then a hyperplane crossing F is G–essential if and only if it is G0 –essential,
so FbG , FbG0 cross the same hyperplanes, i.e. they are parallel. If G0 ≤ G, then FbG is G0 –invariant,
and if [G : G0 ] < ∞, then G0 also acts cocompactly on F , so we can take FbG = FbG0 .


ON HIERARCHICAL HYPERBOLICITY OF CUBICAL GROUPS

10

1.2. Hyperclosure and factor systems.
Definition 1.14 (Factor system, hyperclosure). The hyperclosure of X is the intersection F of
all sets F0 of convex subcomplexes of X that satisfy the following three properties:
(1) X ∈ F0 , and for all combinatorial hyperplanes H of X , we have H ∈ F0 ;
(2) if F, F 0 ∈ F0 , then gF (F 0 ) ∈ F0 ;
(3) if F ∈ F0 and F 0 is parallel to F , then F 0 ∈ F0 .
Note that F is Aut(X )–invariant. If there exists ξ such that for all x ∈ X , there are at most ξ
elements F ∈ F with x ∈ F , then, following [BHS14], we call F a factor system for X .
Remark 1.15. The definition of a factor system in [BHS14] is more general than the definition
given above. The assertion that X has a factor system in the sense of [BHS14] is equivalent to
the assertion that the hyperclosure of X has finite multiplicity, because any factor system (in
the sense of [BHS14]) contains all elements of F whose diameters exceed a given fixed threshold.
Each of the five conditions in Definition 8.1 of [BHS14] is satisfied by F, except possibly the
finite multiplicity condition, Definition 8.1.(3). Indeed, parts (1),(2),(4) of that definition are
included in Definition 1.14 above. Part (5) asserts that there is a constant p so that gF (F 0 ) is in
the factor system provided F, F 0 are and diam(gF (F 0 )) ≥ p. Hence Definition 1.14.(2) implies
that this condition is satisfied by F, with p = 0.
2. Analysis of the hyperclosure
Fix a proper X with a group G acting properly and cocompactly. Let F be the hyperclosure.
2.1. Decomposition. Let F0 = {X } and, for each n ≥ 1, let Fn be the subset of F consisting
of those subcomplexes that can be written in the form gH (F ), where F ∈ Fn−1 and H is a
combinatorial hyperplane. Hence F1 is the set of combinatorial hyperplanes in X .
Lemma 2.1 (Decomposing F). Each F ∈ F − {X } is parallel to a subcomplex of the form
gH1 (gH2 (· · · gHn−1 (Hn ) · · · ))
for some n ≥ 1, where each Hi is a combinatorial hyperplane, i.e. F/k = (∪n≥1 Fn ) /k .
Proof. This follows by induction, Lemma 1.8, and the definition of F.



Corollary 2.2. F = ∪n≥0 Fn .
Proof. It suffices to show F ⊆ ∪n≥0 Fn . Let F ∈ F. If F = X , then F ∈ F0 . Otherwise,
by Lemma 2.1, there exists n ≥ 1, a combinatorial hyperplane H, and a convex subcomplex
F 0 ∈ ∪k≤n Fk with F k gH (F 0 ). Consider φF (PF ∼
= F × F ⊥ ) and choose f ∈ F ⊥ so that
φF (F × {f }) coincides with F . Then φF (F × {f }) lies in some combinatorial hyperplane H 0
– either H 0 = H and F = gH (F ), or F is non-unique in its parallelism class, so lies in a
combinatorial hyperplane in the carrier of a hyperplane crossing F ⊥ . Consider gH 0 (gH (F 0 )).
On one hand, gH 0 (gH (F 0 )) ∈ ∪k≤n+1 Fk . But gH 0 (gH (F 0 )) = F . Hence F ∈ ∪n≥1 Fn .

2.2. Stabilizers act cocompactly. The goal of this subsection is to prove that StabG (F ) acts
cocompactly on F for each F ∈ F. The following lemma is standard but we include a proof in
the interest of a self-contained exposition.
Lemma 2.3 (Coboundedness from finite multiplicity). Let X be a metric space and let G →
Isom(X) act cocompactly, and let Y be a G–invariant collection of subspaces such that every ball
intersects finitely many elements of Y. Then StabG (P ) acts coboundedly on P for every P ∈ Y.

ON HIERARCHICAL HYPERBOLICITY OF CUBICAL GROUPS

11

Proof. Let P ∈ Y, choose a basepoint r ∈ X, and use cocompactness to choose t < ∞ so that
d(x, G · r) ≤ t for all x ∈ X. Choose g1 , . . . , gs ∈ G so that the G–translates of P intersecting
N10t (r) are exactly gi P for i ≤ s. Since Y is G–invariant and locally finite, s < ∞. (In
other words, the assumptions guarantee that there are finitely many cosets of StabG (P ) whose
corresponding translates of P intersect N10t (r), and we have fixed a representative of each of
these cosets.) Let Kr = maxi≤s d(r, gi r). For each g ∈ G, the translates of P that lie within
distance 10t of g · r are precisely gg1 P, . . . , ggs P and Kgr = Kr since d(r, gi · r) = d(g · r, ggi · r).
Fix a basepoint p ∈ P and let q ∈ P be an arbitrary point; choose hp , hq ∈ G so that
d(hp · r, p) ≤ t, d(hq · r, q) ≤ t. Without loss of generality, we may assume that hq = 1. Then
{hp gi P }si=1 is the set of P –translates intersecting N10t (hp · r). Now, p ∈ P and d(hp · r, p) < 10t,
so there exists i so that hp gi P = P , i.e. hp gi ∈ StabG (P ). Finally,
d(hp gi · q, p) ≤ d(hp gi · r, hp gi · q) + d(hp · r, p) + d(hp gi · r, hp · r) ≤ 2t + Kr ,
which is uniformly bounded. Hence the action of StabG (P ) on P is cobounded.



Remark 2.4. We use Lemma 2.3 when X and P are proper, to get a cocompact action.
Lemma 2.5. Let X be a proper CAT(0) cube complex with a group G acting cocompactly. Let
Y, Y 0 ⊂ X be parallel convex subcomplexes, then StabG (Y ) and StabG (Y 0 ) are commensurable.
Thus, if StabG (Y ) acts cocompactly on Y , then StabG (Y ) ∩ StabG (Y 0 ) acts cocompactly on Y 0 .
Proof. Let T be the set of StabG (Y )–translates of Y 0 . Then each gY 0 ∈ T is parallel to Y , and
dX (gY 0 , Y ) = dX (Y 0 , Y ). Since Y ⊥ is locally finite, |T | < ∞. Hence K = ker(StabG (Y ) →
Sym(T )) has finite index in StabG (Y ) but lies in StabG (Y ) ∩ StabG (Y 0 ). By Lemma 1.7, K acts
cocompactly on Hull(Y ∪ Y 0 ), stabilizing Y 0 , and thus acts cocompactly on Y 0 .

Definition 2.6. Let H ∈ F1 . For n ≥ 1, k ≥ 0, let Fn,H,k be the set of F ∈ Fn so that F =
gH (F 0 ) for some F 0 ∈ Fn−1 with d(H, F 0 ) ≤ k. Let Fn,H = ∪k≥0 Fn,H,k and Fn,k = ∪H∈F1 Fn,H,k .
Proposition 2.7 (Cocompactness). Let n ≥ 1. Then, for any F ∈ Fn , StabG (F ) acts cocompactly on F . Hence StabG (F ) acts cocompactly on F for each F ∈ F.
Proof. The second assertion follows from the first and Corollary 2.2. We argue by double
induction on n, k to prove the first assertion, with k as in Definition 2.6. First, observe that Fn ,
Fn,k are G–invariant for all n, k. Similarly, Fn,H,k is StabG (H)–invariant for all H ∈ F1 .
Base Case: n = 1. From local finiteness of X , cocompactness of the action of G and
Lemma 2.3, we see that StabG (H) acts cocompactly on H for each H ∈ F1 .
Inductive Step 1: (n, k) for all k implies (n + 1, 0). Let F ∈ Fn+1,0 . Then F = H ∩ F 0 ,
where H ∈ F1 and F 0 ∈ Fn . By definition, F 0 = gH 0 (F 00 ) for some F 00 ∈ Fn−1 and H 0 ∈ F1 .
Thus K = StabG (F 0 ) acts cocompactly on F 0 by induction.
Let S = {k(H ∩ F 0 ) : k ∈ K}, which is a K–invariant set of convex subcomplexes of F 0 .
Moreover, since the set of all K–translates of H is a locally finite collection, because X is locally
finite and H is a combinatorial hyperplane, S has the property that every ball in F 0 intersects
finitely many elements of S. Lemma 2.3, applied to the cocompact action of K on F 0 , shows
that StabK (H ∩ F 0 ) (which equals StabK (F )), and hence StabG (F ), acts cocompactly on F .
Inductive Step 2: (n, k) implies (n, k + 1). Let F ∈ Fn,k+1 so that F = gH (F 0 ) with
H ∈ F1 , F 0 ∈ Fn−1 and d = d(H, F 0 ) ≤ k + 1. If d ≤ k, induction applies. Thus, we can
assume that d = k + 1. Then there is a product region F × [0, d] ⊂ X with F × {0} = F ,
and F × {d} ⊂ F 0 . Then F1 := F × {1} is a parallel copy of F , and F1 ⊂ gH 0 (F 0 ) with
d(H 0 , F 0 ) = d − 1 ≤ k. By induction L = StabG (gH 0 (F 0 )) acts cocompactly on gH 0 (F 0 ).
We claim that F1 = gH 0 (F 0 ) ∩ gH 0 (H). To see this, note that the hyperplanes that cross F1
are exactly the hyperplanes that cross F 0 and H. However, those are the hyperplanes which
cross H 0 and F 0 which also cross H. It easily follows that the two subcomplexes are equal.

ON HIERARCHICAL HYPERBOLICITY OF CUBICAL GROUPS

12

Now let T be the set of L–translates of F1 = gH 0 (F 0 ) ∩ gH 0 (H) in gH 0 (F 0 ). This is an L–
invariant collection of convex subcomplexes of gH 0 (F 0 ). Moreover, each ball in gH 0 (F 0 ) intersects
finitely many elements of T . Indeed, T is a collection of subcomplexes of the form T` =
g`H 0 (`H) ∩ gH 0 (F 0 ), where ` ∈ L. Recall that dX (H, H 0 ) = 1. Hence, fixing y ∈ gH 0 (F 0 )
and t ≥ 0, if {T`i }i∈I ⊆ T is a collection of elements of T , all of which intersect Nt (y), then
{`i H, `i H 0 }i∈I all intersect Nt+1 (y). However, by local finiteness of X there are only finitely
many distinct `i H, `i H 0 . Further, if `i H = `j H and `i H 0 = `j H 0 , then T`i = T`j . Thus, the
index set I must be finite. Hence, by Lemma 2.3 and cocompactness of the action of L on
gH 0 (F 0 ), we see (as in Inductive Step 1) that StabG (F1 ) acts cocompactly on F1 . Now, since F1
is parallel to F , we see by Lemma 2.5 that StabG (F ) acts cocompactly on F .

The next Lemma explains how to turn the algebraic conditions on the G–action described
in Section 4 into geometric properties of the convex subcomplexes in F. This is of independent
interest, giving a complete algebraic characterization of when two cocompact subcomplexes have
parallel essential cores.
Lemma 2.8 (Characterization of commensurable stabilizers). Let Y1 and Y2 be two convex
subcomplexes of X and let Gi = StabG (Yi ). Suppose further that Gi acts on Yi cocompactly.
c1 and the G2 –essential
Then G1 and G2 are commensurable if and only of the G1 –essential core Y
c
core Y2 are parallel.
c1 , Y
c2 are parallel, then Lemma 2.5 shows that StabG (Y
c1 ), StabG (Y
c2 ) contain
Proof. First, if Y
c
c
b
StabG (Y1 ) ∩ StabG (Y2 ) as a finite-index subgroup. Since StabG (Yi ) contains Gi as a finite-index
subgroup, it follows that G1 ∩ G2 has finite index in G1 and in G2 .
Conversely, suppose that G1 , G2 have a common finite-index subgroup. Thus, G1 ∩ G2 acts
cocompactly on both Y1 and Y2 . This implies that Y1 , Y2 lie at finite Hausdorff distance, since
choosing r > 0 and yi ∈ Yi so that (G1 ∩ G2 )Br (yi ) = Yi , we see that Y1 is in the d(y1 , y2 ) + r
c1 , Y
c2 lie at finite Hausdorff
neighbourhood of Y2 , and vice-versa. Further, this implies that Y
distance, since Ybi is finite Hausdorff distance from Yi .
c1 , Y
c2 are not parallel. Then, without loss of generality, some hyperplane H of
Suppose that Y
c1 but not Y
c2 . Since G1 acts on Y
c1 essentially and cocompactly, [CS11] provides a
X crosses Y
←
− ←
−
←
−
c
hyperbolic isometry g ∈ G1 of Y1 so that g H ( H , where H is the halfspace of X associated
to H and disjoint from Y2 . Choosing n > 0 so that the translation length of g n exceeds the
distance from Y2 to the point in which some g–axis intersects H, we see that H cannot separate
←
−
g n Yb2 from the axis of g. Thus, g n Fb0 ∩ H 6= ∅, whence hgi ∩ G2 = {1}, contradicting that G1
and G2 are commensurable (since g has infinite order). Thus Yb1 , Yb2 are parallel.

2.3. Ascending or descending chains. We reduce Theorem A to a claim about chains in F.
T
Lemma 2.9 (Finding chains). Let U ⊆ F be an infinite subset satisfying U ∈U U 3 x for some
x ∈ X . Then one of the following holds:
• there exists a sequence {Fi }i≥1 in F so that x ∈ Fi ( Fi+1 for all i;
• there exists a sequence {Fi }i≥1 in F so that x ∈ Fi and Fi ) Fi+1 for all i.
Proof. Let Fx ⊇ U be the set of F ∈ F with x ∈ F . Let Ω be the directed graph with vertex set
Fx , with (F, F 0 ) a directed edge if F ( F 0 and there does not exist F 00 ∈ Fx with F ( F 00 ( F 0 .
Let F0 = {x}. Since x is the intersection of the finitely many hyperplane carriers containing
it, F0 ∈ F and in particular F0 ∈ Fx . Moreover, note that F0 has no incoming Ω–edges, since
F0 cannot properly contain any other subcomplex. For any F ∈ Fx , either Ω contains an edge
from F0 to F , or there exists F 0 ∈ Fx such that F0 ⊂ F 0 ⊂ F .

ON HIERARCHICAL HYPERBOLICITY OF CUBICAL GROUPS

13

Hence either Fx contains an infinite ascending or descending ⊆–chain, or Ω is a connected
directed graph in which every non-minimal vertex as an immediate predecessor, and every nonmaximal vertex has an immediate successor. In the first two cases, we are done, so assume that
the third holds. In the third case, there is a unique vertex namely F0 , with no incoming edges
and there is a finite length directed path from F0 to any vertex.
Let F ∈ Fx and suppose that {Fi }i is the set of vertices of Ω so that (F, Fi ) is an edge. For
i 6= j, we have F ⊆ Fi ∩ Fj ( Fi , so since Fi ∩ Fj = gFi (Fj ) ∈ F, we have Fi ∩ Fj = F .
The set {Fi }i is invariant under the action of StabG (F ). Also, by Proposition 2.7, StabG (F )
acts cocompactly on F . A 0–cube y ∈ F is diplomatic if there exists i so that y is joined to a
vertex of Fi − F by a 1–cube in Fi . Only uniformly finitely many Fi can witness the diplomacy
of y since X is uniformly locally finite and Fi ∩ Fj = F whenever i 6= j. Also, y is diplomatic,
witnessed by Fi1 , . . . , Fik , if and only if gy is diplomatic, witnessed by gFi1 , . . . , gFik , for each
g ∈ StabG (F ). Since StabG (F ) y F cocompactly, we thus get |{Fi }i / StabG (F )| < ∞.
b be the graph with a vertex for each F ∈ F containing a point of G · x and a directed
Let Ω
b is a graded directed graph as above. For each
edge for minimal containment as above. Then Ω
b at distance n from a minimal element. The above
n ≥ 0, let Sn be the set of vertices in Ω
b
argument shows that G acts cofinitely on each Sn , and thus Ω/G
is locally finite. Hence, by
b
b (0) /G is finite. In the former
König’s infinity lemma, either Ω/G
contains a directed ray or Ω
b must contain a directed ray, in which case there exists {Fi } ⊆ F with Fi ( Fi+1 for all
case, Ω
i. Up to translating by an appropriate element of G, we can assume that x ∈ F1 . The latter
case means that the set of F ∈ F such that F ∩ G · x 6= ∅ is G–finite. But since G acts properly
and cocompactly on X , any G–invariant G–finite collection of subcomplexes whose stabilizers
act cocompactly has finite multiplicity, a contradiction.

3. Orthogonal complements of compact sets and the hyperclosure
We now characterise F in a CAT(0) cube complex X , without making use of a group action.
Lemma 3.1. Let A ⊆ B ⊆ X be convex subcomplexes and let a ∈ A. Then φB ({a} × B ⊥ ) ⊆
φA ({a} × A⊥ ).
Proof. Let x ∈ φB ({a} × B ⊥ ). Then every hyperplane H separating x from a separates two
parallel copies of B and thus separates two parallel copies of A, since A ⊆ B. It follows
from Lemma 1.11 that every hyperplane separating a from x crosses φA ({a} × A⊥ ), whence
x ∈ φA ({a} × A⊥ ).

⊥
Given a convex subcomplex F ⊆ X , fix a base 0–cube f ∈ F and
 for brevity,
 let F =
⊥
⊥
φF ({f } × F ⊥ ) ⊆ X . Note that f ∈ F ⊥ , and so we let F ⊥ = φF ⊥ {f } × (F ⊥ )
(here, the

(F ⊥ )

⊥

is the abstract orthogonal complement of F ⊥ ), which again contains f , and so we can

 ⊥ ⊥
similarly define F ⊥
etc.
Lemma 3.2. Let F be a convex subcomplex of X . Then



F⊥

 ⊥ ⊥

= F ⊥.

Proof. If F is a convex subcomplex, there is a parallel copy of F ⊥ based at each 0–cube of
⊥
F , since F × F ⊥ is a convex subcomplex of X . Thus F ,→k F ⊥ , and by Lemma 3.1 we

 ⊥ ⊥
have F ⊥ ⊇ F ⊥
. To obtain the other inclusion, we show that every parallel copy of F
⊥
is contained in a parallel copy of F ⊥ . This is clear since, letting A = F ⊥ , we have that
φA (A × A⊥ ) is a convex subcomplex of X , but F ⊂ A⊥ by the above, and thus φF (F × F ⊥ ) ⊆

ON HIERARCHICAL HYPERBOLICITY OF CUBICAL GROUPS

14


⊥
 ⊥ ⊥
φF ⊥ (F ⊥ × F ⊥ ), both of which are convex subcomplexes of X . Hence F ⊥ ⊆ F ⊥
,
completing the proof.

3.1. Characterisation of F using orthogonal complements of compact sets. In this
section, we assume that X is locally finite, but do not need a group action.
Theorem 3.3. Let F ⊂ X be a convex subcomplex. Then F ∈ F if and only if there exists a
compact convex subcomplex C so that C ⊥ = F .
Proof. Let C be a compact convex subcomplex of X . Let H1 , . . . , Hk be the hyperplanes crossing
C. Fix a basepoint x ∈ C, and suppose the Hi are labeled so that x ∈ N (Hi ) for 1 ≤ i ≤ m,
T
and x 6∈ N (Hi ) for i > m, for some m ≤ k. Let F = ki=1 gH1 (Hi ), which contains x. Any
hyperplane H crosses φC ({x} × C ⊥ ) if and only if H crosses each Hi , which occurs if and only
if H crosses F . Hence F = φC ({x} × C ⊥ ), as required.
We now prove the converse. Let F ∈ Fn for n ≥ 1. If n = 1 and F is a combinatorial
hyperplane, F = e⊥ for some 1–cube e of X . Next, assume that n ≥ 2 and write F = gH (F 0 )
where F 0 ∈ Fn−1 and H is a combinatorial hyperplane. Induction on n gives F 0 = (C 0 )⊥ for
some compact convex subcomplex C 0 .
Let e be a 1–cube with orthogonal complement H ∈ F1 , chosen as close as possible to C 0 ,
so that d(e, C 0 ) = d(H, C 0 ). In particular, any hyperplane separating e from C 0 separates H
from C 0 . Moreover, we can and shall assume that C 0 was chosen in its parallelism class so that
d(e, C 0 ) is minimal when e, C 0 are allowed to vary in their parallelism classes.
Let C be the convex hull of (the possibly disconnected set) e ∪ C 0 .
We claim that gH (F 0 ) = {x} × C ⊥ . First, suppose that V is a hyperplane crossing {x} × C ⊥ .
Then V separates two parallel copies of C, each of which contains a parallel copy of e and one
of C 0 . Hence V crosses H and F 0 , so V crosses gH (F 0 ). Thus {x} × C ⊥ ⊆ gH (F 0 ).
Conversely, suppose V is a hyperplane crossing gH (F 0 ), i.e. crossing H and F 0 . To show
that V crosses {x} × C ⊥ , it suffices to show that V crosses every hyperplane crossing C. If W
crosses C, then either W separates e, C 0 or crosses e ∪ C 0 . In the latter case, V crosses W since
V crosses H and (C 0 )⊥ = F 0 . In the former case, since e, C 0 are as close as possible in their
parallelism classes, W separates e, C 0 only if it separates H from C 0 × (C 0 )⊥ , so W must cross
V . Hence gH (F 0 ) ⊆ {x} × C ⊥ . Since only finitely many hyperplanes V either cross e, cross C 0 ,
or separate e from C 0 , the subcomplex C is compact.

⊥

Corollary 3.4. If F ∈ F, then (F ⊥ ) = F .
⊥ ⊥

Proof. If F ∈ F, then F = C ⊥ for some compact C, by Theorem 3.3, and hence ((C ⊥ ) ) =
⊥
(F ⊥ ) = C ⊥ = F , by Lemma 3.2.

4. Auxiliary conditions
In this section, the group G acts geometrically on the proper CAT(0) cube complex X .
4.1. Rotation.
Definition 4.1 (Rotational). The action of G on X is rotational if the following holds. For
each hyperplane B, there is a finite-index subgroup KB ≤ StabG (B) so that for all hyperplanes
A with d(A, B) > 0, and all k ∈ KB , the carriers N (A) and N (kA) are either equal or disjoint.
Remark 4.2. For example, if G\X is (virtually) special, then G acts rotationally on X , but
one can easily make examples of non-cospecial rotational actions on CAT(0) cube complexes.
To illustrate how to apply rotation, we first prove a lemma about F2 .

ON HIERARCHICAL HYPERBOLICITY OF CUBICAL GROUPS

15

Lemma 4.3 (Uniform cocompactness in F2 under rotational actions). Let G act properly, cocompactly, and rotationally on X . Then for any ball Q in X , there exists s ≥ 0, depending only
on X , and the radius of Q, so that for all A, B ∈ F1 , at most s distinct translates of gB (A) can
intersect Q.
Proof. Note that if B, gB are in the same G–orbit, and KB ≤ StabG (B) witnesses the rotation
g
of the action at B, then KB
does the same for gB, so we can assume that the index of KB ∈
StabG (B) is uniformly bounded by some constant ι as B varies over the (finitely many orbits
of) combinatorial hyperplanes.
Next, note that it suffices to prove the claim for Q of radius 0, since the general statement
will then follow from uniform properness of X .
Finally, it suffices to fix combinatorial hyperplanes B and A and bound the number of
StabG (B)–translates of A whose projections on B contain some fixed 0–cube x ∈ B, since
only boundedly many translates of B can contain x.
We can assume that A is disjoint from B, for otherwise gB (A) = A ∩ B, and the number of
translates of A containing x is bounded in terms of G and X only.
Now suppose d(A, B) > 0. First, let {g1 , . . . , gk } ⊂ KB be such that the translates gi gB (A)
T
T
are all distinct and x ∈ ki=1 gB (gi A) = ki=1 gi gB (A). For simplicity, we can and shall assume
that g1 = 1.
We can also assume, by multiplying our eventual bound by 2, that the gi A all lie on the same
side of B, i.e. the hyperplane B 0 whose carrier is bounded by B and a parallel copy of B does
not separate any pair of the gi A. By rotation, A, g2 A are disjoint, and hence separated by some
hyperplane V .
Since V cannot separate gB (A), gB (g2 A), we have that V separates either A or g2 A from
B. (The other possibility is that V = B but we have ruled this out above.) Up to relabelling,
we can assume the former. Then, for i ≥ 2, we have that gi V separates gi A from gi B = B.
Moreover, by choosing V as close as possible to B among hyperplanes that separate A from B
and g2 A, we see that the hyperplanes {gi V }ki=1 have pairwise-intersecting carriers, and at least
two of them are distinct. This contradicts the rotation hypothesis unless k = 1.
More generally, the above argument shows that if {g1 , . . . , gk } ⊂ StabG (B) are such that the
T
T
translates gi gB (A) are all distinct and x ∈ ki=1 gB (gi A) = ki=1 gi gB (A), then the number of gi
belonging to any given left coset of KB in StabG (B) is uniformly bounded. Since [StabG (B) :
KB ] ≤ ι, the lemma follows.

More generally:
Lemma 4.4. Let G act properly, cocompactly, and rotationally on X . Then there exists a
constant s0 so that the following holds. Let F ∈ F. Then for all ρ ≥ 0, at most s0 distinct
G–translates of F can intersect any ρ–ball in X .
Proof. As in the proof of Lemma 4.3, it suffices to bound the number of G–translates of F
containing a given 0–cube x. As in the same proof, it suffices to bound the number of StabG (B)–
translates of F containing x, where B is a combinatorial hyperplane for which x ∈ F ⊂ B.
By the first paragraph of the proof
Tn of Theorem 3.3, there exists n and combinatorial
Tn−1 hyperplanes A1 , . . . , An such that F = i=1 gB (Ai ). If An is parallel to B, then F = i=1 gB (Ai ),
so by choosing a smallest such collection, we have that no Ai is parallel to B.
Let gT1 , . . . , gk ∈ KB have the property that the gi F are all distinct and contain x. Note that
gi F = nj=1 gB (gi Aj ).
Now, if Aj is disjoint from B, then rotation implies that for all i, either gi Aj = Aj or
gi Aj ∩ Aj are disjoint. If gi Aj 6= Aj , there is a hyperplane V separating them, and V cannot
cross or coincide with B (as in the proof of Lemma 4.3). Hence V separates Aj , say, from B.
So gi V separates Aj from B, and gi V 6= gj V . By choosing V as close as possible to B, we have

ON HIERARCHICAL HYPERBOLICITY OF CUBICAL GROUPS

16

(again as in Lemma 4.3) that V and gi V cross or osculate, which contradicts rotation. Hence
gi Aj = Aj for all such i, j.
T
Let J be the set of j ≤ n so that Aj is disjoint from B, so that j∈J gB (Aj ) is fixed by each
gi . Let J 0 = {1, 2, . . . , n} − J. Note that for all j ∈ J 0 , the combinatorial hyperplane Aj is one
of at most χ combinatorial hyperplanes that contain x and χ is the maximal degree of a vertex
in X . Moreover, gB (ATj ) = Aj ∩ Bj .
Since each gi fixes j∈J gB (Aj ), k must be bounded in terms of the number of translates of
T
j∈J 0 gB (Aj ) containing x; since we can assume that the Aj contain x, this follows.
As in the proof of Lemma 4.3, if g1 , . . . , gk ∈ StabG (B) have the property that the gi F are all
distinct and contain x, then the number of gi belonging to any particular hKB , h ∈ StabG (B)
is uniformly bounded, and the number of such cosets is bounded by ι, so the number of such
StabG (B)–translates of F containing x is uniformly bounded.

We now prove Theorem A in the special case where G acts on X rotationally.
Corollary 4.5. Let G act properly, cocompactly, and rotationally on the proper CAT(0) cube
complex X . Then F is a factor system.
Proof. By Lemma 4.4, there exists s0 < ∞ so that for all F ∈ F, at most s0 distinct G–translates
of F can contain a given point. By uniform properness of F and the proof of Lemma 2.3, there
exists R < ∞ so that each F ∈ F has the following property: fix a basepoint x ∈ F . Then for
any y ∈ F , there exists g ∈ StabG (F ) so that d(gx, y) ≤ R. Hence there exists k so that for all
F , the complex F contains at most k StabG (F )–orbits of cubes.
Conclusion in the virtually torsion-free case: If G is virtually torsion-free, then (passing
to a finite-index torsion-free subgroup) G\X is a compact nonpositively-curved cube complex
admitting a local isometry StabG (F )\F → G\X , where StabG (F )\F is a nonpositively-curved
cube complex with at most k cubes. Since there are only finitely many such complexes, and
finitely many such local isometries, the quotient G\F is finite. Since each x ∈ S is contained
in boundedly many translates of each F ∈ F, and there are only finitely many orbits in F, it
follows that x is contained in boundedly many elements of F, as required.
General case: Even if G is not virtually torsion-free, we can argue essentially as above, except we have to work with nonpositively-curved orbi-complexes instead of nonpositively-curved
cube complexes.
First, let Y be the first barycentric subdivision of X , so that G acts properly and cocompactly
on Y and, for each cell y of Y, we have that StabG (y) fixes y pointwise (see [BH99, Chapter
III.C.2].) Letting F 0 be the first barycentric subdivision of F , we see that F 0 is a subcomplex of
Y with the same properties with respect to the StabG (F )–action. Moreover, F 0 has at most k 0
StabG (F )–orbits of cells, where k 0 depends on dim X and k, but not on F .
The quotient G\Y is a complex of groups whose cells are labelled by finitely many different
finite subgroups, and the same is true for StabG (F )\F . Moreover, we have a morphism of
complexes of groups StabG (F )\F → G\Y which is injective on local groups. Since G acts
on X properly, the local groups in G\Y are finite. Hence there are boundedly many cells in
StabG (F )\F , each of which has boundedly many possible local groups (namely, the various
subgroups of the local groups for the cells of G\Y). Hence there are finitely many choices of
StabG (F )\F , and thus finitely many G–orbits in F, and we can conclude as above.

4.2. Weak height and essential index conditions.
Definition 4.6 (Weak height condition). Let G be a group and H ≤ G a subgroup. The
subgroup H satisfies theTweak height condition if the following holds. Let {gi }i∈I be an infinite
subset of G so that H ∩ i∈J H gi is infinite whenever J ⊂ I is a finite subset. Then there exists
i, j so that H ∩ H gi = H ∩ H gj .

ON HIERARCHICAL HYPERBOLICITY OF CUBICAL GROUPS

17

Definition 4.7 (Noetherian Intersection of Conjugates Condition (NICC)). Let G be a group
and H ≤ G a subgroup. The subgroup H satisfies the Noetherian intersection of conjugates
condition (NICC) if the
holds. Let {gi }∞
i=1 be an infinite subset of distinct elements of
Tn following
g
i
G so that Hn = H ∩ i=1 H is infinite for all n, then there exists ` > 0 so that for all j, k ≥ `,
Hj and Hk are commensurable.
Definition 4.8 (Conditions for hyperplanes). Let G act on the CAT(0) cube complex X .
Then the action satisfies the weak height condition for hyperplanes or respectively NICC for
hyperplanes if, for each hyperplane B of X , the subgroup StabG (B) ≤ G satisfies the weak
height condition, or NICC, respectively.
Remark 4.9. Recall that H ≤ G has finite height if there exists n so that any collection of at
least n + 1 distinct left cosets of H has the property that the intersection of the corresponding
conjugates of H is finite. Observe that if H has finite height, then it satisfies both the weak
height condition and NICC, but that the converse does not hold.
Definition 4.10 (Essential index condition). The action of G on X satisfies the essential index
condition if there exists ζ ∈ N so that for all F ∈ F we have [StabG (Fb) : StabG (F )] ≤ ζ, where
Fb is the StabG (F )–essential core of F .
4.3. Some examples where the auxiliary conditions are satisfied. We now briefly consider some examples illustrating the various hypotheses. Our goal here is just to illustrate the
conditions in simple cases.
4.3.1. Special groups. Stabilizers of hyperplanes in a right-angled Artin groups are simply special
subgroups generated by the links of vertices. Let Γ be a graph generating a right angled Artin
group AΓ and let Λ be any inducted subgraph. Then AΛ is a special subgroup of AΓ , and AgΛi
g
has non-trivial intersection with AΛj if and only if gi gj−1 commutes with some subgraph Λ0 of Λ.
Further, their intersection is conjugate to the special subgroup AΛ0 . The weak height condition
and NICC follow. The essential index condition also holds since each AΛ acts essentially on the
corresponding element of F, which is just a copy of the universal cover of the Salvetti complex of
AΛ . In fact, these considerations show that hyperplane stabilisers in RAAGs have finite height.
It is easily verified that these properties are inherited by subgroups arising from compact local
isometries to the Salvetti complex, reconfirming that (virtually) compact special cube complexes
have factor systems in their universal covers.
4.3.2. Non-virtually special lattices in products of trees. The uniform lattices in products of trees
from [Wis96, BM97, Rat07, JW09] do not satisfy the weak height condition, but they do satisfy
NICC and the essential index condition.
Indeed, let G be a cocompact lattice in Aut(T1 × T2 ), where T1 , T2 are locally finite trees. If
A, B are disjoint hyperplanes, then gB (A) is a parallel copy of some Ti , i.e. gB (A) is again a
hyperplane; otherwise, if A, B cross, then gB (A) is a single point. The essential index condition
follows immediately, as does the NICC. However, G can be chosen so that there are pairs of
parallel hyperplanes A, B so that StabG (A) ∩ StabG (B) has arbitrarily large (finite) index in
StabG (B), so the weak height condition fails.
4.3.3. Graphs of groups. Let Γ be a finite graph of groups, where each vertex group Gv acts
properly and cocompactly on a CAT(0) cube complex Xv with a factor system Fv , and each edge
group Ge acts properly and cocompactly on a CAT(0) cube complex Xe , with a factor system
Fe , so that the following conditions are satisfied, where v, w are the vertices of e:
• there are G–equivariant convex embeddings Xe → Xv , Xw
• these embedding induce injective maps Fe → Fv , Fw .

ON HIERARCHICAL HYPERBOLICITY OF CUBICAL GROUPS

18

If the action of G on the Bass-Serre tree is acylindrical, then one can argue essentially as in the
proof of [BHS15, Theorem 8.6] to prove that the resulting tree of CAT(0) cube complexes has a
factor system. Moreover, ongoing work on improving [BHS15, Theorem 8.6] indicates that one
can probably obtain the same conclusion in this setting without this acylindricity hypothesis.
Of course, one can imagine gluing along convex cocompact subcomplexes that don’t belong
to the factor systems of the incident vertex groups. Also, we believe that the property of being
cocompactly cubulated with a factor is preserved by taking graph products, and that one can
prove this by induction on the size of the graph by splitting along link subgroups. This is the
subject of current work in the hierarchically hyperbolic setting.
4.3.4. Cubical small-cancellation quotients. There are various ways of building more exotic examples of non-virtually special cocompactly cubulated groups using groups. In [Hua16, JW17],
Jankiewicz-Wise construct a group G that is cocompactly cubulated but does not virtually
split. They start with a group G0 of the type discussed in Remark 4.2 and consider a smallcancellation quotient of the free product of several copies of G0 . This turns out to satisfy strong
cubical small-cancellation conditions sufficient to produce a proper, cocompact action of G on
a CAT(0) cube complex. However, it appears that the small-cancellation conditions needed
to achieve this are also strong enough to ensure that the NICC and essential index properties
pass from G0 to G. The key points are that G is hyperbolic relative to G0 , and each wall in G
intersects each coset of G0 in at most a single wall (Lemma 4.2 and Corollary 4.5 of [JW17]).
5. F is closed under orthogonal complementation, given a group action
We now assume that X is a locally finite CAT(0) cube complex on which the group G acts
properly and cocompactly. Let F be the hyperclosure in X and let B be a constant so that each
0–cube x of X lies in ≤ B combinatorial hyperplanes.
For convex subcomplexes D, F of X , we write F = D⊥ to mean F = φD ({f } × D⊥ ) for some
f ∈ D, though we may abuse notation, suppress the φD , and write e.g. {f } × D⊥ to mean
φD ({f } × D⊥ ) when we care about the specific point f .
Proposition 5.1 (F is closed under orthogonal complements). Let G act on X properly and
cocompactly. Suppose that one of the following holds:
• the G–action on X satisfies the weak height property for hyperplanes;
• the G–action on X satisfies the essential index condition and the NICC for hyperplanes;
• F is a factor system.
Let A be a convex subcomplex of X . Then A⊥ ∈ F. Hence StabG (A⊥ ) acts on A⊥ cocompactly.
In particular, for all F ∈ F, we have that F ⊥ ∈ F.
We first need a lemma.
Lemma 5.2. Let A ⊂ X be a convex subcomplex with diam(A) > 0 and let x ∈ A(0) . Let
H1 , . . . , Hk be all of the hyperplanes intersecting A whose carriers contain x, so that for each i,
there is a combinatorial hyperplanes Hi+ associated to Hi with x ∈ Hi+ . Let Y = ∩ki=1 Hi+ . Let
S be the set of all combinatorial hyperplanes associated to hyperplanes crossing A. Then
\
A⊥ =
gY (H 0 ),
H 0 ∈S

where A⊥ denotes the orthogonal complement of A at x. In particular, if A is unique in its
parallelism class, then A⊥ = {x}. Finally, if diam(A) = 0, then A⊥ = X .
Proof. If A is a single 0–cube, then A⊥ = X by definition. Hence suppose that diam(A) > 0.
Let H 0 ∈ S. Since gY (H 0 ∩ A) ⊆ Y ∩ A = {x}, we see that x ∈ gY (H 0 ). Suppose that y ∈ A⊥ .
Then every hyperplane V separating y from x crosses each of the hyperplanes H 0 crossing A, and

ON HIERARCHICAL HYPERBOLICITY OF CUBICAL GROUPS

19

T
thus crosses Y , whence yT∈ gY (H 0 ) for each H 0 ∈ S. Thus A⊥ ⊆ H 0 ∈S gY (H 0 ). On the other
hand, suppose that y ∈ H 0 ∈S gY (H 0 ). Then every hyperplane H 0 separating x from y crosses
T
every hyperplane crossing A, so y ∈ A⊥ . This completes the proof that A⊥ = H 0 ∈S gY (H 0 ).
Finally, A is unique in its parallelism class if and only if A⊥ = {x}, by definition of A⊥ . 
We can now prove the proposition:
Proof of Proposition 5.1. The proof has several stages.
Setup using Lemma 5.2: If A is a single point, then A⊥ = X , which is in F by definition.
Hence suppose diam(A) > 0, and let H1 , . . . , Hk , x ∈ A, Y ⊂ X , and S be as in Lemma 5.2, so
\
A⊥ =
gY (H 0 ).
H 0 ∈S

Thus, to prove the proposition, it is sufficient to produce a finite collection H of hyperplanes
H 0 crossing A so that
\
\
gY (H 0 ) =
gY (H 0 ).
H 0 ∈S

H 0 ∈H

Indeed, if there is such a collection, then we have shown A⊥ to be the intersection of finitely
many elements of Fk , whence A⊥ ∈ Fk+|H| , as required. Hence suppose for a contradiction that
for any finite collection H ⊂ S, we have
\
\
gY (H 0 ) (
gY (H 0 ).
H 0 ∈S

H 0 ∈H

Bad hyperplanes crossing Y : For each m, let Hm be the (finite) set of hyperplanes H 0
A (x) = A ∩ N (x) (and hence satisfying x ∈ g (H 0 )).
intersecting Nm
m
Y
Consider the collection Bm of all hyperplanes W such that W crosses each element of Hm and
W crosses Y , but W fails to cross A⊥ . (This means that there exists j > m and some U ∈ Hj
so that W ∩ U = ∅.)
Suppose that there exists m so that Bn = ∅ for n > m. Then we can take Hm to be our
desired set H, and we are done. Hence suppose that Bm is nonempty for arbitrarily large m.
A (x) is parallel into U , but there exists j > m so that N A (x) is
Note that if U ∈ Bm , then Nm
j
A
not parallel into U . (Here Nj (x) denotes the j–ball in A about x.)
Elements of Bm osculating A⊥ : Suppose that U ∈ Bm , so that NjA (x) is not parallel into
U for some j > m. Suppose that U 0 is a hyperplane separating U from A⊥ .
Then U 0 separates U from x so, since U intersects Y and x ∈ Y , we have that U 0 intersects Y .
Since gY (A) = {x} is not crossed by any hyperplanes, U 0 cannot cross A. Hence U 0 separates
A (x), so N A (x) is parallel into U 0 (since it is parallel into U ). On the other hand,
U from Nm
m
0
since U separates U from A⊥ , U 0 cannot cross A⊥ , and thus fails to cross some hyperplane
crossing A. Hence U 0 ∈ Bm . Thus, for each m, there exists Um ∈ Bm whose carrier intersects
A⊥ . Indeed, we have shown that any element of Bm as close as possible to A⊥ has this property.
Hence we have a sequence of radii rn and hyperplanes Un so that:
• Un crosses Y ;
• N (Un ) ∩ A⊥ 6= ∅;
• NrAn (x) is parallel into Un for all n;
• NrAn+1 (x) is not parallel into Un , for all n.
The above provides a sequence {Vn } of hyperplanes so that for each n:
• Vn crosses A;
• Vn crosses Um for m ≥ n;
• Vn does not cross Um for m < n.

ON HIERARCHICAL HYPERBOLICITY OF CUBICAL GROUPS

20

Indeed, for each n, choose Vn to be a hyperplane crossing NrAn+1 (x) but not crossing Un . For
each n, let V̄n be one of the two combinatorial hyperplanes (parallel to Vn ) bounding N (Vn ).
Claim 1. For each n, the subcomplex A⊥ is parallel into V̄n .
Proof of Claim 1. Let H be a hyperplane crossing A⊥ . Then, by definition of A⊥ , H crosses
each hyperplane crossing A. But Vn crosses A, so H must also cross Vn . Thus A⊥ is parallel
into Vn .

Next, since G acts on X cocompactly, it acts with finitely many orbits of hyperplanes, so,
by passing to a subsequence (but keeping our notation), we can assume that there exists a
hyperplane V crossing A and elements gn ∈ G, n ≥ 1 so that Vn = gn V for n ≥ 1. For
simplicity, we can assume g1 = 1.
Next, we can assume, after moving the basepoint x ∈ A a single time, that x ∈ V̄ , i.e. V̄ is
among the k combinatorial hyperplanes whose intersection is Y . This assumption is justified by
the fact that F is closed under parallelism, so it suffices to prove that any given parallel copy of
A⊥ lies in F. Thus we can and shall assume Y ⊂ V̄ .
For each m, consider the inductively defined subcomplexes Z1 = V̄ and for each m ≥ 2,
Zm = gV̄ (gg1 V̄ (· · · (ggm−1 V̄ (gm V̄ )) · · · )), which is an element of F.
Claim 2. For all m ≥ 1, we have A⊥ ⊆ Zm .
Proof of Claim 2. Indeed, since A⊥ ⊂ Y by definition, and Y ⊂ V̄ , we have A⊥ ⊂ V̄ . On the
other hand, for each n, Claim 1 implies that A⊥ is parallel into gn V̄ for all n, so by induction,
A⊥ ⊂ Zm for all m ≥ 1, as required.

Claim 3. For all m ≥ 1, we have Zm ) Zm+1 .
Proof of Claim 3. For each m, the hyperplane Um crosses Y , by construction. Since Y ⊆ V̄ ,
this implies that Um crosses V̄ . On the other hand, Um does not cross V̄m+1 . This implies that
Zm 6= Zm+1 . On the other hand, Zm+1 ⊂ Zm just by definition.

Tm
Let Km = StabG (V ) ∩ n=1 StabG (V )gn ; by Lemma 1.9 Km has finite index in StabG (Zm ).
Thus, since Zm ∈ F, we see that Km acts on Zm cocompactly. Claim 3 implies that no Zm is
compact, for otherwise we would be forced to have Zm = Zm+1 for some m. Since Km acts on
Zm cocompactly, it follows that Km is infinite for all m.
Thus far, we have not used any of the auxiliary hypotheses. We now explain how to derive a
contradiction under the weak finite height hypothesis.
Claim 4. Suppose that the G–action on X satisfies weak finite height for hyperplanes. Then,
after passing to a subsequence, we have Km = K2 for all m ≥ 2.
T
Proof of Claim 4. Let I ⊂ N be a finite set and let m = max I. Then n∈I StabG (V )gi contains
Km , and is thus infinite, since Km was shown above to be infinite. Hence, since StabG (V ) satisfies the weak finite height property, there exist distinct m, m0 so that StabG (V ) ∩ StabG (V )gm =
0
StabG (V ) ∩ StabG (V )gm .
0
Declare m ∼ m0 if StabG (V ) ∩ StabG (V )gm = StabG (V ) ∩ StabG (V )gm , so that ∼ is an
equivalence relation on N. If any ∼–class [m] is infinite, then we can pass to the subsequence
0
[m] and assume that StabG (V ) ∩ StabG (V )gn = StabG (V ) ∩ StabG (V )gn for all n. Otherwise, if every ∼–class is finite, then there are infinitely many ∼–classes, and we can pass to
a subsequence containing one element from each ∼–class. This amounts to assuming that
0
StabG (V ) ∩ StabG (V )gm 6= StabG (V ) ∩ StabG (V )gm for all distinct m, m0 , but this contradicts
weak finite height, as shown above.
Hence, passing to a subsequence, we can assume that StabG (V ) ∩ StabG (V )gm = StabG (V ) ∩
0
StabG (V )gm for all m, m0 ≥ 2, and hence Km = K2 for all m.


ON HIERARCHICAL HYPERBOLICITY OF CUBICAL GROUPS

21

From Claim 4 and the fact that Km stabilises Zm for each m, we have that K2 stabilises each
Zm . Moreover, K2 acts on ZTm cocompactly. T
Now, by Claim 2, A⊥ ∈ m≥1 Zm , so x ∈ m≥1 Zm since x ∈ A⊥ . The fact that K2 acts
on Z2 cocompactly provides a constant R so that each point of Z2 lies R–close to the orbit
K2 · x. In fact, for any m ≥ 2, the K2 –invariance of Zm and the fact that x ∈ Zm implies that
Zm ⊂ NR (K2 · x).
Recall that for each m, the hyperplane Um has the property that d(Um , A⊥ ) ≤ 1, and hence
d(Um , Zm0 ) ≤ 1 for all m, m0 , by Claim 2. Thus d(Um , K2 · x) ≤ R + 1 for all m. So, there exists
an increasing sequence (mi )i so that the hyperplanes Umi all belong to the same K2 –orbit, since
only boundedly many hyperplanes intersect any (R + 1)–ball. Hence there is a single hyperplane
U and, for each i, an element ki ∈ K2 so that Umi = ki U .
Now, since K2 = Kmi for all i, we have that K2 stabilises V and each of the hyperplanes
gmi V . In particular, for each i, the set of hyperplanes crossing V and gmi V is K2 –invariant,
and the same is true for the set of hyperplanes crossing V but failing to cross gmi V . However,
our construction of the set of Um implies that there exists i0 so that U crosses Vmi0 but does
not cross Vmi0 +1 . Thus the same is true for each K2 –translate of U and hence for each Umi .
This contradicts our original construction of the Um ; this contradiction shows that the claimed
sequence of Um cannot exist, whence A⊥ ∈ F.
Having proved the proposition under the weak finite height assumption, we now turn to the
other hypotheses. Let {Zm } and {Km } be as above.
Claim 5. Suppose that the G–action on X satisfies the NICC and the essential index condition.
Then there exists ` so that StabG (Zm ) = StabG (Z` ) for all m ≥ `.
T
Proof of Claim 5. Let I ⊂ N be a finite set and let m = max I. Then n∈I StabG (V )gi contains
Km , and is thus infinite, since Km was shown above to be infinite. Hence, since StabG (V )
satisfies the NICC, there exists ` so that Km is commensurable with K` for all m ≥ `.
bm and Zb` are parallel for all m ≥ `. Moreover, since
Hence, by Lemma 2.8, we have that Z
Km ≤ K` , Proposition 1.13 implies that we can choose essential cores within their parallelism
classes so that Zbm = Zb` for all m ≥ `.
b` ) = StabG (Zbm ) for all m ≥ `. The essential index condition implies that
Let L = StabG (Z
StabG (Zm ) has uniformly bounded index in L as m → ∞, so by passing to a further infinite
subsequence, we can assume that StabG (Zm ) = StabG (Z` ) for all m ≥ ` (since L has finitely
many subgroups of each finite index).

Claim 5 implies that (up to passing to a subsequence), StabG (Z` ) = StabG (Zm ) preserves
Zm (and acts cocompactly on Zm ) for all m ≥ `.
For simplicity, we pass to a finite-index subgroup Γ of StabG (Z` ) = StabG (Zm ) that preserves
V (and necessarily still acts cocompactly on each Zm ). Such a subgroup exists since Zm is
contained in only finitely many hyperplanes, and that set of hyperplanes must be preserved
under the action of StabG (Z` ).
Claim 6. Let ` ≤ m < m0 . Then for all g ∈ Γ, we have that Um crosses gV and gVm but does
not cross Vm0 or gVm0 . Hence, for all g ∈ Γ, the hyperplane gUm crosses V and Vm but not Vm0 .
Proof of Claim 6. By definition, Zm and gVm (Zm ) are parallel. Hence gZm and ggVm (gZm ) are
parallel. But g ∈ Γ ≤ StabG (Vm ), so gZm = Zm . Hence ggVm (Zm ) is parallel to Zm . So, a
hyperplane U crosses Zm if and only if U crosses ggVm (Zm ). Now, Um crosses V and Vm , and
thus crosses Zm and hence crosses ggVm (Zm ). Thus Um crosses gVm .
On the other hand, suppose Um crosses gVm0 . Then since Um crosses V (because it crosses
Zm ), we have that Um crosses g −1 V = V and Vm0 , which contradicts our choice of the Um and
Vm0 when m < m0 .

ON HIERARCHICAL HYPERBOLICITY OF CUBICAL GROUPS

22

To verify the final assertion, observe that by the first assertion, Um crosses V = g −1 V and
−1 ∈ Γ, so gU crosses V and V . On the other hand, U does not cross g −1 V 0 ,
m , since g
m
m
m
m
so gUm cannot cross Vm0 .

g −1 V

We can now argue exactly as in the weak finite height argument, with L playing the role
of K2 , to obtain the same contradiction and conclude that A⊥ ∈ F. Specifically, just using
Γ–cocompactness of Z` , we can pass to a subsequence of the Um that all belong to a single
Γ–orbit. These hyperplanes Um all cross V , and the set of m0 such that Um crosses Vm0 is
independent of which Um in the Γ–orbit we have chosen, by Claim 6. As before, this contradicts
our construction of the sequences (Um ), (Vm ), and hence A⊥ ∈ F.
Applying the factor system assumption: If F is a factor system, then since Zm ∈ F for
all m, and Zm ) Zm+1 for all m, we have an immediate contradiction, so A⊥ ∈ F.
Conclusion: We have shown that under any of the additional hypotheses, A⊥ ∈ F when
A ⊂ X is a convex subcomplex. This holds in particular if A ∈ F.

The preceding proposition combines with earlier facts to yield:
Corollary 5.3 (Ascending and descending chains). Let G act properly and cocompactly on X ,
satisfying any of the hypotheses of Proposition 5.1. Suppose that for all N ≥ ∞, there exists a 0–
cube x ∈ X so that x lies in at least N elements of F. Then there exist sequences (Fi )i≥1 , (Fi0 )i≥1
of subcomplexes in F so that all of the following hold for all i ≥ 1:
• Fi ( Fi+1 ;
0 ;
• Fi0 ) Fi+1
• Fi0 = Fi⊥ .
Moreover, there exists a 0–cube x that lies in each Fi and each Fi0 .
Proof. Lemma 2.9, cocompactness, and G–invariance of F provide a sequence (Fi ) in F and a
point x so that x ∈ Fi for all i and either Fi ( Fi+1 for all i, or Fi ) Fi+1 for all i. For each i,
let Fi0 = φFi ({x} × Fi⊥ ). Proposition 5.1 implies that each Fi0 ∈ F, and Lemma 3.1 implies that
(Fi0 ) is an ascending or descending chain according to whether (Fi ) was descending or ascending.
⊥ = F 0 , then by Corollary 3.4,
Assume first that Fi ( Fi+1 for all i. Now, if Fi0 = Fi⊥ = Fi+1
i+1
0
0
we have Fi = Fi+1 , a contradiction. Hence (Fi ) is properly descending, i.e. Fi0 ) Fi+1
for all i.
The case where (Fi ) is descending is identical. This completes the proof.

6. Proof of Theorem A
We first establish the setup. Recall that X is a proper CAT(0) cube complex with a proper,
cocompact action by a group G. We denote the hyperclosure by F; our goal is to prove that
there exists N < ∞ so that each 0–cube of X is contained in at most N elements of F, under
any of the three additional hypotheses of Theorem A.
If there is no such N , then Corollary 5.3 implies that there exists a 0–cube x ∈ X and a
sequence (Fi )i≥1 in F so that x ∈ Fi ( Fi+1 for each i ≥ 1. For the sake of brevity, given any
subcomplex E 3 x, let E ⊥ denote the orthogonal complement of E based at x. Corollary 5.3
⊥ for all i. Proposition 2.7 shows that Stab (F ⊥ )
also says that Fi⊥ ∈ F for all i and Fi⊥ ) Fi+1
G i
acts on Fi⊥ cocompactly
for
all
i.
S
T
Let U = i Fi and let I = i Fi⊥ , and note that U ⊥ = I and I ⊥ = U . In particular,
⊥
⊥
U ⊥ = I ⊥ = U and I ⊥ = U ⊥ = I. From here, we can now prove our main theorem:
Proof of Theorem A. We have already proved the theorem under the rotation hypothesis, in
Corollary 4.5. Hence suppose that either weak finite height holds or the NICC and essential
index conditions both hold, so that Proposition 5.1 implies that U, I ∈ F.

ON HIERARCHICAL HYPERBOLICITY OF CUBICAL GROUPS

23

By Corollary 3.4, we have compact convex subcomplexes D, E with U = D⊥ and I = E ⊥ .
Moreover, we can take D ⊂ I and E ⊂ U . Now, Corollary 3.4, Theorem 3.3, and Proposition 5.1
provide, for each i ≥ 1, a compact, convex subcomplex Ci , containing x and contained in Fi , so
that Ci⊥ = Fi⊥ .
0
Let C10 = C1 and, for each i ≥ 2, let Ci0 = Hull(Ci−1
∪ Ci ). Then Ci0 is convex, by definition,
and compact, being the convex hull of the union of a pair of compact subspaces. Now, Ci0 ⊆ Fi
0
for each i. Indeed, C10 = C1 ⊆ F1 by construction. Now, by induction, Ci−1
⊂ Fi−1 , so
0
0
Ci−1 ⊂ Fi since Fi−1 ⊂ Fi . But Ci ⊂ Fi by construction, so Ci ∪ Ci−1 ⊂ Fi , whence the hull of
that union, namely Ci0 , lies in Fi since Fi is convex. Hence Ci ⊆ Ci0 ⊆ Fi for i ≥ 1.
By Lemma 3.1, for each i, Fi⊥ ⊆ (Ci0 )⊥ ⊆ Ci⊥ = Fi⊥ since Ci ⊆ Ci0 . Hence (Ci0 )i≥1 is an
ascending sequence of convex, compact subcomplexes, containing x, with (Ci0 )⊥ = Fi⊥ for all i.
T
T
S 0 ⊥
S
Note that i (Ci0 )⊥ = i Fi⊥ = I. However, I = ( S
Ci ) . But, Ci0 cannot be compact since
0 , and by Corollary 3.4, we can choose E ⊆
0 for some R. But by
Ci0 ( Ci+1
Ci0 . Thus E ⊆ CR
⊥
⊥
⊥
0
Lemma 3.1 this means that I = E ⊇ (CR ) = FR , a contradiction. Thus, F must have finite
multiplicity, as desired.

We show now that for cube complexes that admit geometric actions, having a factor system
implies the NICC for hyperplanes and essential index conditions for hyperplanes. Thus, any
proof that F forms a factor system for all cocompact cubical groups must necessarily show that
any group acting geometrically on a CAT(0) cube complex satisfies these conditions.
Theorem 6.1. Let X be a CAT(0) cube complex admitting a geometric action by group G. If
F is a factor system, then G satisfies NICC for hyperplanes and the essential index condition.
Proof. Suppose that F is a factor system and at most N elements of F can contain any given
x ∈ X (0) . Then for any A, B ∈ F1 , there are at most N elements of F which can contain Fb, the
StabG (F )–essential core of F . In particular, there are at most N distinct StabG (Fb)–translates
of F . Thus, [StabG (Fb), StabG (F )] ≤ N , verifying the essential index condition.
To verify NICC for hyperplanes, let H be a hyperplane and let K = Stab
(H). Let {gi }∞
i=1
TG
n
be sequence of distinct elements of G so that for n ≥ 1, the subgroup K ∩ i=1 K gi is infinite.
Consider the hyperplane H, notice that K gi is the stabilizer of gi H. Now, consider F1 =
gH (g1 H) and inductively define Fk = gFk−1 (gH (gk H)). Since F is a factor system, the set of
G–translates of Fk−1 and gH (gk H) have finite multiplicity for all k ≥ 2, and so we can apply
the argument of Lemma 1.9 and induction to conclude that StabG (Fk ) is commensurable with
T
Gk = K ∩ ki=1 K gk , which is infinite by assumption.
Since F is a factor system, there must be some ` so that for all k ≥ `, Fk = F` . In this case,
Gk and G` are commensurable for all k ≥ `, and in particular Gk and Gk0 are commensurable
for all k, k 0 ≥ `, and thus G satisfies NICC for hyperplanes.

7. Factor systems and the simplicial boundary
Corollary D follows from Theorem A, Proposition 7.1 and [Hag13, Lemma 3.32]. Specifically,
the first two statements provide a combinatorial geodesic ray representing each boundary simplex
v, and when v is a 0–simplex, [Hag13, Lemma 3.32] allows one to convert the combinatorial
geodesic ray into a CAT(0) ray. Proposition 7.1 is implicit in the proof of [DHS16, Theorem
10.1]; we give a streamlined proof here.
Proposition 7.1. Let X be a CAT(0) cube complex with a factor system F. Then each simplex σ
of ∂4 X is visible, i.e. there exists a combinatorial geodesic ray α such that the set of hyperplanes
intersecting α is a boundary set representing the simplex σ.
Remark 7.2. Proposition 7.1 does not assume anything about group actions on X , but instead
shows that the existence of an invisible boundary simplex is an obstruction to the existence of

ON HIERARCHICAL HYPERBOLICITY OF CUBICAL GROUPS

24

a factor system. The converse does not hold: counterexamples can be constructed by beginning
with a single combinatorial ray, and gluing to the nth vertex a finite staircase Sn , along a single
vertex. The staircase Sn is obtained from [0, n]2 by deleting all squares that are strictly above
the diagonal joining (0, 0) to (n, n). In this case, F has unbounded multiplicity, and any factor
system must contain all elements of F exceeding some fixed threshold diameter, so the complex
cannot have a factor system.
Proof of Proposition 7.1. Let σ be a simplex of ∂4 X . Let σ 0 be a maximal simplex containing
σ, spanned by v0 , . . . , vd . The existence of σ 0 follows from [Hag13, Theorem 3.14], which says
that maximal simplices exist since X is finite-dimensional (otherwise, it could not have a factor
system). By Theorem 3.19 of [Hag13], which says that maximal simplices are visible, σ 0 is
visible, i.e. there exists a combinatorial geodesic ray γ such that the set H(γ) of hyperplanes
crossing γ is a boundary set representing σ 0 . We will prove that each 0–simplex vi is visible. It
then follows from [Hag13, Theorem 3.23] that any face of σ 0 (hence σ) is visible.
Let Y be the convex hull of γ. The set of hyperplanes crossing Y is exactly H(γ). Since
Y is convex in X , Lemma 8.4 of [BHS14], which provides an induced factor system on convex
subcomplexes of cube complexes with factor systems, implies that Y contains a factor system.
F
By Theorem 3.10 of [Hag13], we can write H(γ) = di=1 Vi , where each Vi is a minimal
boundary set representing the 0–simplex vi . Moreover, up to reordering and discarding finitely
many hyperplanes (i.e. moving the basepoint of γ) if necessary, whenever i < j, each hyperplane
H ∈ Vj crosses all but finitely many of the hyperplanes in Vi .
For each 1 ≤ i ≤ d, minimality of Vi provides a sequence of hyperplanes (Vni )n≥0 in Vi so
i
that Vni separates Vn±1
for n ≥ 1 and so that any other U ∈ Vi separates Vmi , Vni for some m, n,
by the proof of [Hag13, Lemma 3.7] or [CFI16, Lemma B.6] (one may have to discard finitely
many hyperplanes from Vi for this to hold; this replaces γ with a sub-ray and shrinks Y).
We will show that, after discarding finitely many hyperplanes from H(γ) if necessary, every
element of Vi crosses every element of Vj , whenever i 6= j. Since every element of Vi either lies
in (Vni )n or separates two elements of that sequence, it follows that U and V cross whenever
T
U ∈ Vi , V ∈ Vj and i 6= j. Then, for any i, choose n ≥ 0 and let H = j6=i Vnj . Projecting γ to
H yields a geodesic ray in Y, all but finitely many of whose dual hyperplanes belong to Vi , as
required. Hence it suffices to show that Vni and Vmj cross for all m, n whenever i 6= j.
j
Fix j ≤ d and i < j. For each n ≥ 0, let m(n) ≥ 0 be minimal so that Vm(n)
fails to cross Vni .
Note that we may assume that this is defined: if Vni crosses all Vmj , then, since Vmj crosses all
but finitely many of the hyperplanes from Vi , it crosses Vki for k >> n. Since it also crosses Vni ,
it must also cross Vri for all n ≤ r ≤ k. By discarding Vki for k ≤ n we complete the proof. Now
suppose that m(n) is bounded as n → ∞. Then there exists N so that Vni , Vmj cross whenever
m, n ≥ N , and we are done, as before.
Hence suppose that m(n) → ∞ as n → ∞. In other words, for all m ≥ 0, there exists
n ≥ 0 so that Vmj crosses Vki if and only if k ≥ n. Choose M  0 and choose n maximal with
j
m(n) < M . Then all of the hyperplanes Vm(k)
with k ≤ n cross Vki , . . . , Vni but do not cross Vti
j
for t < k. Hence the subcomplexes gVni (Vki ), k ≤ n are all different: gVni (Vki ) intersects Vm(k)
but
i
i
i
i
gVni (Vk−1 ) does not. On the other hand, since Vk separates V` from Vn when ` < k < n, every
hyperplane crossing Vni and V`i crosses Vki , so gVni (Vki ) ∩ gVni (V`i ) 6= ∅. Thus the factor system on
Y has multiplicity at least n. But since m(n) → ∞, we could choose n arbitrarily large in the
preceding argument, violating the definition of a factor system.


Proof of Corollary E. If γ is a CAT(0) geodesic, then it can be approximated, up to Hausdorff
distance depending on dim X , by a combinatorial geodesic, so assume that γ is a combinatorial
geodesic ray. By Corollary D, the simplex of ∂4 X represented by γ is spanned by 0–simplices

ON HIERARCHICAL HYPERBOLICITY OF CUBICAL GROUPS

25

v0 , . . . , vd with each vi represented by a Q
combinatorial geodesic ray γi . Theorem 3.23 of [Hag13]
0
0
0
says
that
X
contains
a
cubical
orthant
i γi , where each γi represents vi . Hence Hull(∪i γi ) =
Q
0
i Hull(γi ). Up to truncating an initial subpath of γ, we have that γ is parallel into Hull(∪i γi )
(and thus lies in a finite neighbourhood of it). The projection of the original CAT(0) geodesic
approximated by γ to each Hull(γi0 ) is a CAT(0) geodesic representing vi . The product of these
geodesics is a combinatorially isometrically embedded (d + 1)-dimensional orthant subcomplex
of Y containing (the truncated) CAT(0) geodesic in a regular neighbourhood.

In the presence of a proper, cocompact group action, we can achieve full visibility under
slightly weaker conditions than those that we have shown suffice to obtain a factor system:
Proposition 7.3. Let X be a proper CAT(0) cube complex on which the group G acts properly
and cocompactly. Suppose that the action of G on X satisfies NICC for hyperplanes. Then each
simplex σ of ∂4 X is visible, i.e. there exists a combinatorial geodesic ray α such that the set of
hyperplanes intersecting α is a boundary set representing the simplex σ.
Proof. We adopt the same notation as in the proof of Proposition 7.1. As in that proof, if ∂4 X
contains an invisible simplex, then we have two infinite sets {Vi }i≥0 , {Hj }j≥0 of hyperplanes
with the following properties:
• for each i ≥ 1, the hyperplane Hi separates Hi−1 from Hi+1 ;
• for each j ≥ 1, the hyperplane Vj separates Vj−1 from Vj+1 ;
• there is an increasing sequence (ij ) so that for all j, Vj crosses Hi if and only if i ≤ ij .
This implies that for all i ≥ 1, the subcomplex Fi = gH0 (gH1 (· · · (gHi−1 (Hi )) · · · )) is unbounded.
Since StabG (Fi ) acts cocompactly, by Proposition 2.7, StabG (Fi ) is infinite. By Lemma 1.9,
T
StabG (Fi ) is commensurable with Ki = ij=1 StabG (Hj ), and so by NICC, there exists N so
that Ki is commensurable with KN for all i ≥ N . Thus, after passing to a subsequence, we
see that for all i, the Ki –essential core of Fi is a fixed nonempty (indeed, unbounded) convex
subcomplex Fb of H0 .
Now, for each j, the hyperplane Vj cannot cross Fb, because Fb lies in Fi for all i, and Vj fails
to cross Hi for all sufficiently large i. Moreover, this shows that Fb must lie in the halfspace
associated to Vj that contains Vj+1 . But since this holds for all j, we have that Fb is contained
in an infinite descending chain of halfspaces, contradicting that Fb 6= ∅.

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DPMMS, University of Cambridge, Cambridge, UK
Current address: School of Mathematics, University of Bristol, Bristol, UK
E-mail address: markfhagen@gmail.com
Mathematics Department, Bard College at Simon’s Rock, Great Barrington, Massachusetts,
USA
E-mail address: tisusse@gmail.com

