Hyperbolic groups acting improperly

In this paper we study hyperbolic groups acting on CAT(0) cube complexes. The first main result (Theorem A) is a structural result about the Sageev construction, in which we relate quasi-convexity of hyperplane stabilizers with quasi-convexity of cell stabilizers. The second main result (Theorem D) generalizes both Agol's theorem on cubulated hyperbolic groups and Wise's Quasi-convex Hierarchy Theorem.


Introduction
In recent years, CAT(0) cube complexes have played a central role in many spectacular advances, most notably in Agol's proof of the Virtual Haken Conjecture [1]. A fundamental way in which cube complexes arise is via a construction of Sageev [24] which takes as input a group G and a collection of codimension-1 subgroups of G and produces a CAT(0) cube complex X, equipped with an isometric G-action on X with no global fixed point.
Sageev's construction works in great generality. However, in order to get more information from the G-action on X, it is useful to add geometric hypotheses. For example, if G is a hyperbolic group and the codimension-1 subgroups are quasi-convex, Sageev proved that the associated cube complex is G-cocompact [25,Theorem 3.1]. Achieving a proper action is harder (see [5,21] for conditions which ensure properness).
Even an improper action G X gives a description of G as the fundamental group of a complex of groups in the sense of Bridson-Haefliger (see [8,III.C]

or
The first author was partially supported by the Simons Foundation, #342049 to Daniel Groves and the National Science Foundation, DMS-1507067. The second author was also partially supported by the Simons Foundation #524176 to Jason Manning and the National Science Foundation, DMS-1462263. Section 2 below). In this description, the underlying space is G X and the local groups can be identified with cell stabilizers for the action.
Our first main result links the geometry of the hyperplane stabilizers with that of the cell stabilizers.
Theorem A. Let G be hyperbolic. The following conditions on a cocompact Gaction on a CAT(0) cube complex are equivalent: (1) All hyperplane stabilizers are quasi-convex.
Intersections of quasi-convex subgroups are quasi-convex, and cell stabilizers are intersections of vertex stabilizers. Therefore, the equivalence of (2) and (3) is trivial. We prove the equivalence of (1) and (2).
We remark that we actually prove the direction (1) =⇒ (2) in the more general setting of arbitrary finitely generated groups where we assume the relevant subgroups are strongly quasi-convex in the sense of [26]. Note that in this more general setting, (2) and (3) are still equivalent. See Section 3 for more details. In Subsection 3.7 we explain how Theorem A implies the following result.
Corollary B. Suppose that G is a hyperbolic group acting cocompactly on a CAT(0) cube complex X with quasi-convex hyperplane stabilizers. Then (1) X is δ-hyperbolic for some δ; (2) there exists a k ≥ 0 so that the fixed point set of any infinite subgroup of G intersects at most k distinct cells; and (3) the action of G on X is acylindrical (in the sense of Bowditch [6, p. 284]).
Anthony Genevois explained to us how conclusion (2) implies acylindricity for actions on hyperbolic CAT(0) cube complexes (see Subsection 3.7). The condition in (2) is not implied by acylindricity since X is not assumed to be locally compact.
Without the conclusion of δ-hyperbolicity, a more general version of Corollary B holds just as for Theorem A. See Remark 3.44 for more details.
In Sageev's construction, the stabilizers in G of hyperplanes in the resulting cube complex are commensurable with the chosen codimension-1 subgroups of G. Therefore, we have the following result.
Corollary C. Let G be a hyperbolic group and let H = {H 1 , . . . , H k } be a collection of quasi-convex codimension-1 subgroups. Let X be a CAT(0) cube complex obtained by applying the Sageev construction to H.
(1) The stabilizers of cells in X are quasi-convex in G. In particular, they are finitely presented. (2) X is δ-hyperbolic for some δ.
(3) There exists a k ≥ 0 so that the fixed set of any infinite subgroup of G intersects at most k distinct cells. (4) The action of G on X is acylindrical.
As far as we are aware, even the corollary of item (1) that the cell stabilizers are finitely generated in the above result is new. We remark that the fact that cell stabilizers are finitely presented implies that the description of G as the fundamental group of the complex of groups associated to G X is a finite description.
Some of the most dramatic uses of CAT(0) cube complexes have come from Haglund and Wise's theory of special cube complexes [18]. A cube complex is special if it admits a locally isometric immersion into the Salvetti complex of a right-angled Artin group. A group G is virtually special if there is a finite-index subgroup G 0 ≤ G and a CAT(0) cube complex X so that G 0 acts freely and cubically on X and G 0 X is a compact special cube complex. (For some authors the quotient is allowed to be non-compact but have finitely many hyperplanes.) As shown in [18], virtually special hyperbolic groups have many remarkable properties, such as being residually finite, linear over Z and possessing very strong subgroup separability properties.
Agol [1] proved that if a hyperbolic group G acts properly and cocompactly on a CAT(0) cube complex then G is virtually special. It is this result that implies the Virtual Haken Conjecture, as well as the Virtual Fibering Conjecture (in the compact case), and many other results.
One of the key ingredients of the proof of Agol's Theorem, and another of the most important theorems in the area is Wise's Quasi-convex Hierarchy Theorem [27,Theorem 13.3] (see also [3,Theorem 10.2]) which states that if a hyperbolic group G can be expressed as A * C (respectively A * C B) where C is quasi-convex in G and A is (respectively A and B are) virtually special then G is virtually special. This theorem can be rephrased as saying that if a hyperbolic group acts cocompactly on a 1-dimensional CAT(0) cube complex (otherwise known as a 'tree') with virtually special and quasi-convex cell stabilizers, then G is virtually special.
Our second main result is a common generalization of Agol's theorem and Wise's Quasi-convex Hierarchy Theorem.
Theorem D. Suppose that G is a hyperbolic group acting cocompactly on a CAT(0) cube complex X so that cell stabilizers are quasi-convex and virtually special. Then G is virtually special.
By Corollary C, Theorem D has the following immediate consequence.
Corollary E. Suppose that G is a hyperbolic group and that H = {H 1 , . . . , H k } is a collection of quasi-convex codimension-1 subgroups. If the vertex stabilizers of the G-action on a cube complex obtained by applying the Sageev construction to H are virtually special, then G is virtually special.
Since finding proper actions of hyperbolic groups on CAT(0) cube complexes is much harder than finding cocompact actions, Theorem D is expected to be a powerful new tool for proving that hyperbolic groups are virtually special. Already Yen Duong [11] has used Theorem D to show that random groups in the square model at density < 1/3 are virtually special.
Theorem A is one of the key ingredients of the proof of Theorem D. We now explain how Theorem D is a consequence of the above-mentioned results of Agol and Wise, along with Theorem A and the following result (proved in Section 6).
Theorem F. Suppose that the hyperbolic group G acts cocompactly on a CAT(0) cube complex X and that cell stabilizers are virtually special and quasi-convex. There exists a quotient G = G/K so that (1) The quotient K X is a CAT(0) cube complex; (2) The group G is hyperbolic; and (3) The action of G on K X is proper (and cocompact).
Proof of Theorem D. Consider the hyperbolic group G, acting on a CAT(0) cube complex X as in the statement of Theorem D. By Theorem F there exists a hyperbolic quotient G = G/K of G so that K X is a CAT(0) cube complex, and the G-action on K X is proper and cocompact. Let Z = K X . By Agol's Theorem [1,Theorem 1.1], there is a finite-index subgroup G 0 of G so that G 0 Z is special. Let G 0 be the pre-image in G of G 0 . Clearly, the underlying space of G 0 X is the same as that of G 0 Z , and in particular all of the hyperplanes are two-sided and embedded.
We cut successively along these hyperplanes, applying the complex of groups version of the Seifert-van Kampen Theorem [8, Example III.C. 3.11.(5) and Exercise III.C. 3.12]. In this way, we obtain a hierarchy of G 0 with the following properties: (1) The edge groups are quasi-convex (since they are stabilizers of hyperplanes, which are quasi-convex by Theorem A); and (2) The terminal groups are virtually special (since they are finite-index subgroups of the vertex stabilizers in G).
Therefore, G 0 admits a quasi-convex hierarchy terminating in virtually special groups, so G 0 is virtually special by Wise's Quasi-convex Hierarchy Theorem [27,Theorem 13.3] (see [3,Theorem 10.3] for a somewhat different account). Since G 0 is finite-index in G, the group G is virtually special, as required. This completes the proof of Theorem D.
We now briefly outline the contents of this paper. In Section 2 we recall those parts of the theory of complexes of groups from [8] which we need. In Section 3, we prove Theorem A and Corollary B. The proof of Theorem A depends on a quasi-convexity criterion (Theorem A.3) which is proved separately in Appendix A. We separate out Theorem A.3 and its proof both because it may be of independent interest and because the methods, unlike in the rest of the paper, are pure δhyperbolic geometry. In Section 4 we investigate conditions on a group G acting on a CAT(0) cube complex X and a normal subgroup K G so that the quotient K X is a CAT(0) cube complex. In Section 5 we translate these conditions into grouptheoretic statements. In Section 6 we prove various results about Dehn filling (in particular, Theorem 6.4 and Corollary 6.5 which may be of independent interest) to see that the conditions from Section 5 are satisfied for certain subgroups K which arise as kernels of long Dehn filling maps. We use this to deduce Theorem F. 1.1. Notation and conventions. The notation A< B indicates that A is a finite index subgroup of B; similarly, A˙ B indicates A is a finite index normal subgroup. We write conjugation as a x = xax −1 , or sometimes as Ad(x)(a). For p an element of a G-set, we denote the G-orbit by p .
1.2. Acknowledgments. We thank Richard Webb for suggesting that the direction (1) =⇒ (2) of Theorem A might hold in a more general setting than that of hyperbolic groups.
Thanks to Anthony Genevois for pointing out that his work allows us to deduce acylindricity (Corollary B.(3)) from Corollary B. (2).
We also thank Alessandro Sisto for the suggestion of using a result like Theorem 6.6 in our joint work [15]. This result simplifies the proof of Theorem 6.4.

The complex of groups coming from an action on a cube complex
In this section we give a brief account of those parts of the theory of complexes of groups which we need. Much more detail can be found in Bridson-Haefliger [8, III.C].
2.1. Small categories without loops (scwols). By a scwol (small category without loops) we mean a small category in which for every object v, the set of arrows from v to itself contains only the unit 1 v . For an arrow a of a scwol, we denote its source by i(a) and its target by t(a). If i(a) = t(a), we say a is a trivial arrow ; it follows that a = 1 v for some v. We sometimes conflate v and 1 v . A (non-degenerate) morphism of scwols f : A → B is a functor which induces, for each object v of A, a bijection between the arrows {a | i(a) = v} and the arrows Notation 2.1. Given a scwol X , we denote the set of objects of X by V (X ) and the set of non-unit morphisms in X by E(X ). The set E(X ) comes equipped with two maps Where a is a morphism from i(a) to t(a).
Let E ± (X ) be the set of symbols a + and a − as a ranges over E(X ). We refer to elements e of E ± (X ) as oriented edges of X . If e = a + then i(e) = t(a) and t(e) = i(a), while if e = a − then i(e) = i(a) and t(e) = t(a).
A key example of a scwol is the (opposite) poset of cells of a simplicial or cubical complex, with arrows from each cell to all its faces.
Let X be a CAT(0) cube complex, and suppose that G acts on X combinatorially. The quotient G X may or may not be a cube complex, depending on whether the groups G σ = {g | gσ = σ} and {g | gx = x, ∀x ∈ σ} agree for all cells σ.
Another way to phrase this issue is to note that, if X 0 is the scwol of cells of X, then G acts by morphisms on X 0 , but the quotient map X 0 → G X 0 may not be a morphism of scwols, since some isometry of X may fix the center of some cube, but permute faces of that cube. In order to obtain a complex of groups structure on G from the action G X, we need a scwol quotient, so we replace X 0 with X , the scwol of cells of the first barycentric subdivision of X: If W is a cube complex, the idealization of W is a scwol W which has objects V (W) in one-to-one correspondence with non-empty nested chains of cubes of W . There is at most one morphism in W between two objects: If c 1 contains c 2 as a sub-chain, there is an arrow from c 1 to c 2 .
For example, if X is a single 1-dimensional cube e with endpoints a and b, the nontrivial arrows of the idealization X are as follows: Already a square τ with e as a face is much more complicated. The idealization is shown on the left as a graph, with detail shown on the right for the highlighted portion.
Any automorphism of a cube complex gives an automorphism of its idealization. Moreover if φ maps a chain of cubes to itself, then it also preserves all subchains. It follows that the quotient Y = G X is also a scwol, and that the quotient map X → Y is non-degenerate morphism of scwols.

Remark 2.3.
A small category C always has a (geometric) realization which is a simplicial complex whose 0-cells are the objects of C, with 1-cells corresponding to morphisms, 2-cells to composable pairs of morphisms, and so on. The realization of the idealization of W is the second barycentric subdivision of W , so it is naturally homeomorphic to W .

2.2.
Paths and homotopies in a category. The definitions here are mainly taken from [8, III.C.A], though our notation is slightly different.
Let C be a category. We define C-paths to be lists of letters e, where e = a ± for some arrow a of C. For e = a ± we have i(a + ) = t(a) = t(a − ) and t(a + ) = i(a) = i(a − ).
A C-path p of length 0 is an object v of C, with i(p) = t(p) = v. We also consider the path of length 0 at v to be an empty list (though it is an empty list based at v). For j > 0, a C-path of length j is a list p = (e 1 , . . . , e k ) where for each i we have t(e i ) = i(e i+1 ). We have i(p) = i(e 1 ) and t(p) = t(e k ).
If p is a C-path of length j > 0, q is a C-path of length k > 0, and t(p) = i(q), then the concatenation p · q is a C-path of length j + k with i(p · q) = i(p) and t(p · q) = t(q). 1 The category C is connected if for any two If q is any C-path, then i(q) and t(q) can be regarded as paths of length 0. We use the convention that i(q) · q = q · t(q) = q.
Definition 2.5. If C is the realization of a category C, then there is a canonical correspondence between combinatorial paths in the 1-skeleton of C and C-paths.
If p is a combinatorial path in C (1) , and q the corresponding C-path, we say that p is the realization of q, and q is the idealization of p.
Remark 2.6. Suppose that C is the idealization of a cube complex C, so that the realization of C is the second barycentric subdivision of C. In later sections, we make use of the fact that the following types of paths have canonical idealizations in C: (1) Combinatorial paths in the 1-skeleton of the first cubical subdivision C b of C (Section 3). (2) Combinatorial paths in links of cells of C (Section 4). In both cases, this follows from the fact that subdivisions of these graphs embed naturally in the 1-skeleton of the second barycentric subdivision. (1) For each object σ of A, a local group (also called a cell group) H σ ; (2) For each arrow a of A, an injective group homomorphism ψ a : H i(a) → H t(a) (If a is a trivial arrow, we require ψ a to be the identity map); and (3) For each pair of composable arrows a, b with composition a • b, a twisting element z(a, b) ∈ H t(a) . (If either a or b is trivial, z(a, b) = 1.) 2 These data satisfy the following conditions (writing ab for a•b) whenever all written compositions of arrows are defined: (1) (compatibility) Ad(z(a, b))ψ ab = ψ a ψ b ; and (2) (cocycle) ψ a (z(b, c))z(a, bc) = z(a, b)z(ab, c).
Definition 2.8 (The complex of groups coming from an action). Suppose G acts on a scwol X so that any g ∈ G fixing an object fixes every arrow from that object. Let Y be the quotient scwol. We obtain a complex of groups G(Y) once we have made the following choices [8, III.C.2.9]: (1) For each object v of Y, a lift v to X ; this lift also determines lifts a of all arrows a with i(a) = v. (2) For each nontrivial arrow a, a choice of element h a so that t(h a ( a)) = t(a). Given these choices, one defines: ab . The complex of groups G(Y) can be used to recover the group G. There are two different ways of doing this. The first is explained in [8,III.C.3.7], and involves G(Y)-paths. The second way is from [8, III.C.A], and is the way that we proceed. The advantage to this second way, which uses categories and coverings of categories, is that lifting paths to covers is a canonical procedure (as with usual covering theory).

2.4.
Fundamental groups and coverings of categories. In Definition 2.4 we defined homotopy of C-paths, where C is a category. Definition 2.9. Given a category C and an object v 0 of C, the fundamental group of C based at v 0 , denoted π 1 (C, v 0 ), is the set of homotopy classes of C-loops based at v 0 , with operation induced by concatenation of C-paths. 15]. Let C be a connected category. A functor f : C → C is a covering if for each object σ of C the restriction of f to the collection of morphisms that have σ as their initial (respectively, terminal) object is a bijection onto the set of morphisms which have f (σ ) as their initial (resp., terminal) object.
The universal cover C of a connected category C is described in [8,III.C.A.19]: Fix a base vertex v 0 of Y, and define Obj( C) to be the set of homotopy classes of C-paths starting at v 0 . If [c] is a homotopy class of path, and α is an arrow from t(c), then there is an arrow of C from [c] to [c · α − ]. The projection π : C → C sets π ([p]) = t(p) and if α is the arrow described above then π( α) = α.
The theory of coverings of categories is entirely analogous to ordinary covering theory. In fact it is a special case, as the covering spaces of a connected category C correspond bijectively to the covering spaces of its realization.
We record the following observation.
Lemma 2.11. Let φ : C → C be a covering of categories, and suppose φ( v) = v, for objects v of C and v of C. Any C-path p with i(p) = v has a unique lift to a C-path p with i( p) = v. Moreover any elementary homotopy from p to a path p gives a unique elementary homotopy of p to a lift p of p with the same endpoints as p.
2.5. The category associated to a complex of groups. Any complex of groups G(Y) has an associated category CG(Y).
Definition 2.12. [8, III.C.2.8] The objects of CG(Y) are the objects of the scwol Y. Arrows of CG(Y) are pairs (g, a) so that a is an arrow of Y and g ∈ G t(a) . Composition is defined by (g, a) • (h, b) = (gψ a (h)z(a, b), ab).
Recall that if a is a trivial arrow then ψ a is the identity homomorphism and z(a, x), z(x, a) are always trivial.
Remark 2.13. The map CG(Y) → Y given by (g, a) → a is a functor. This map has an obvious section a → (1, a). If there are nontrivial twisting elements this is not a functor, but it does allow Y-paths to be "unscwolified" to CG(Y)-paths. In Definition 2.20, we explain how to go back and forth between paths in covers of CG(Y) and their associated scwols.
Theorem 2.14. [8, III.C.3.15 and III.C.A.13] Suppose that the group G acts on the simply connected complex X, giving rise to an action of G on the scwol X , and that v 0 is an object in Y = G X . Let CG(Y) be the category associated to G(Y).
where v is a vertex of the scwol Y. The arrow (g, 1 v ) is called a group arrow.
In later sections we abuse notation and refer to the edge (g, 1 v ) + (for a group arrow (g, 1 v )) as "(g, v)" or even just "g". We also blur the difference between the scwol arrow (1, a) and the Y-arrow a, and often refer to the scwol arrow by "a". We also blur the distinction between the CG(Y)-edge (1, a) ± and the Y-edge a ± . Lemma 2.16. Every CG(Y)-path is homotopic to a concatenation of group and scwol arrows.
Proof. Observe that any CG(Y)-arrow (g, a) is a composition of a group arrow and a scwol arrow: (g, a) = (g, t(a)) • (1, a).
As described at the end of the last subsection, a choice of base vertex v 0 determines a universal covering map φ : CG(Y) → CG(Y) sending a homotopy class of path [p] to its terminal vertex t(p), and the arrow from [c] to [c · (g, a) − ] to the arrow (g, a) of CG(Y).
What is important for us is that the group π 1 (CG(Y), v 0 ) acts on the universal . An arrow of C is said to be a scwol (resp. group) arrow if its label is a scwol (resp. group) arrow of CG(Y) is any cover, then every C-path is homotopic to a concatenation of group and scwol arrows.
Proof. Lemma 2.16 gives a homotopy in CG(Y) to a path of the desired form. Lemma 2.11 says that the homotopy lifts.
We are particularly interested in covers of CG(Y) corresponding to normal subgroups of G ∼ = π 1 (G(Y), v 0 ). Fix some such K G, let CG(Y) be the universal cover of CG(Y), and let C K := K CG(Y) be the corresponding cover. The group K also acts on the scwol X , with quotient scwol Z := K X . We observe that (just as with ordinary covers) a CG(Y)-loop p represents an element of K if and only if it lifts to a loop in C K . (Since K G, the basepoints do not matter.) Given a regular cover C K of CG(Y), corresponding to the normal subgroup K G, there is a natural quotient C K of C K , defined as follows: Define an equivalence relation on the objects of C K where two objects are equivalent if they differ by an invertible arrow. Define an equivalence relation on arrows of C K by setting γ 1 ∼ γ 2 if i(γ 1 ) ∼ i(γ 2 ), t(γ 1 ) ∼ t(γ 2 ) and if there are invertible arrows: ρ 1 : i(γ 1 ) → i(γ 2 ) and ρ 2 : t(γ 1 ) → t(γ 2 ) so that ρ 2 γ 1 = γ 2 ρ 1 .
Objects and arrows of C K up to the above equivalences form a quotient category C K . Since Mor(v, v) is a group for any object v of C K , the following lemma is straightforward.
Lemma 2.19. The category C K is a scwol, and the quotient map C K → C K is a functor. The category C K is G K -equivariantly isomorphic to the scwol Z = K X . Definition 2.20. We denote the functor from C K to Z given by Lemma 2.19 by Θ K : C K → Z. Since Θ K is a functor, it also gives a way to turn a C K -path p into a Z-pathp. Deleting all the trivial arrows fromp produces a Z-path which we call the scwolification of p. Abusing the notation slightly, we denote the scwolification of p by Θ K (p).
Conversely, if σ is a Z-path, then any C K -path σ so that Θ K ( σ) = σ is called an unscwolification of σ. The unscwolification is highly non-unique, but always exists.
The following can be deduced by examining the elementary homotopies. Given a CG(Y)-path p we can lift it to a C K -path p, and then scwolify p to the Z-path Θ K ( p). In case K = {1} we have Z = X and C K = CG(Y). In this case we just write Θ : CG(Y) → X .

Quasi-convexity in the Sageev construction
In this section, we prove Theorem A. Recall that we have a hyperbolic group G acting cocompactly on a CAT(0) cube complex X, and we are required to prove that the vertex stabilizers are quasi-convex if and only if the hyperplane stabilizers are quasi-convex.
To prepare for this proof it may be useful to think about the case that X is a tree. In that case, hyperplanes are midpoints of edges, and so the statement is that edge stabilizers are quasi-convex if and only if vertex stabilizers are. Edge stabilizers are intersections of vertex stabilizers, and intersections of quasi-convex subgroups are quasi-convex, so one direction is clear. The other direction is not much harder: Consider a geodesic joining two vertices of a vertex stabilizer. The vertex stabilizer is coarsely separated from the rest of the Cayley graph by appropriate cosets of edge stabilizers. The quasi-convexity of these cosets "traps" the geodesic close to the vertex stabilizer. Now remove the assumption that X is a tree, and suppose that vertex stabilizers are quasi-convex. It still follows that edge stabilizers are quasi-convex, but a hyperplane stabilizer is much bigger than an edge stabilizer. We will express a hyperplane stabilizer as a union of cosets of edge stabilizers, intersecting in a controlled way, and use a quasi-convexity criterion proved in the Appendix to conclude that the hyperplane stabilizer is quasi-convex.
If on the other hand we assume that hyperplane stabilizers are quasi-convex, we will use them as in the tree case to control geodesics joining points in a vertex stabilizer. We inductively use more and more hyperplanes to corral points on a geodesic in an argument which terminates because of the finite dimensionality of the cube complex.
As mentioned in the introduction, we prove the direction (1) =⇒ (2) in a more general setting. Therefore, for the beginning of this section we do not assume that G is hyperbolic, merely that it acts cocompactly on a CAT(0) cube complex X.
We briefly describe the contents of the remainder of this section. In Subsection 3.1 we explain how we consider subset of small categories as graphs. In Subsection 3.2 we identify certain subsets of CG(Y) which are tuned to the cubical geometry of X b and associate graphs with these subsets. In Subsection 3.3 we build graphs upon which intersections of stabilizers of hyperplanes act. In Subsection 3.4 we prove the direction (1) =⇒ (2) of Theorem A. In fact, we prove the more general Theorem 3.26. In Subsection 3.5 we prove the direction (2) =⇒ (1) of Theorem A. In Subsection 3.6 we consider various possible generalizations of Theorem A. Finally, in Subsection 3.7 we prove Corollary B.
3.1. Graphs from subsets of small categories. Let C be a (small) category, and let S be a subset of the set of arrows of C. There is an associated graph (really a 1-complex), which we denote Gr(S), with vertex set the set of objects which are either the source or target of some arrow in S, and with edges in correspondence with the arrows S. For Gr(S) a graph constructed this way, we denote the original set of arrows as Ar(Gr(S)) = S. Example 3.2. Suppose C is a group (i.e. C has a single object and each morphism of C is invertible), and S 0 ⊂ C is a generating set. Let S be the set of arrows in the universal cover C with label in S 0 . Then Gr(S) is the Cayley graph of G with respect to S 0 .

Cubical paths.
It is convenient for us to work in the cubical subdivision of X, which we now describe. Definition 3.3. Suppose that X is a cube complex. The cubical subdivision of X, denoted X b , is the cube complex obtained by replacing each n-cube in X by 2 n n-cubes, found by subdividing each coordinate interval into two equal halves, and then gluing in the obvious way induced from the structure of X.
Of course, X b is canonically homothetic to X, and X b is NPC (respectively, CAT(0)) if and only if X is. We suppose that X is CAT(0), and therefore X b is also.
If a group G acts by cubical automorphisms on X, then it clearly does so on X b . Moreover, the carrier of a hyperplane W in X b is homeomorphic to W × [0, 1]. With appropriate choice of orientation, W × {1} is a hyperplane W ↑ of X, and Stab(W ) ⊂ Stab(W ↑ ) is an inclusion with index 1 or 2 (depending on whether or not there is an element of G which fixes W ↑ but exchanges the two sides of W ↑ ). We denote W × {0} by W ↓ . Note that W ↓ naturally corresponds to the cubical subdivision of a sub-complex of X.
We observe the following. Thus, in order to prove Theorem A, we can consider either hyperplane stabilizers for hyperplanes in X or for hyperplanes in X b . A similar result holds in the setting of strongly quasi-convex subgroups of a finitely generated group, as discussed in Subsection 3.4 below.
Observation 3.5. The vertices of X b are in bijection with the cubes of X.
The cells of X b are in bijection with pairs ( σ 1 , σ 2 ) of cubes in X so that σ 1 ⊆ σ 2 . The dimension of the cube corresponding to ( Thus, an edge in X b corresponds to a pair of cubes ( σ 1 , σ 2 ) where σ 1 is a codimension-1 face of σ 2 . Moreover, each cell of X b can be naturally identified with an object of X .
In Section 2.4, we defined CG(Y) to be the universal covering of the category CG(Y) associated to the complex of groups G(Y). Recall that the objects of CG(Y) are homotopy classes of CG(Y)-paths, starting at a (fixed) basepoint v 0 ∈ Y, and arrows are labeled by arrows of CG(Y) (Definition 2.17). It is helpful to assume (as we may do without loss of generality) that v 0 comes from a 0-cube of X. The basepoint of CG(Y) is v 0 , the homotopy class of the constant path at v 0 .
The group π 1 (CG(Y), v 0 ) acts on CG(Y), with quotient the category CG(Y). As in Theorem 2.14, we can identify G with π 1 (CG(Y), v 0 ). The proof of each direction of Theorem A begins with choosing a certain connected G-cocompact subgraph Γ of Gr( CG(Y)). The G-cocompact graph is different in the two directions of the proof, primarily because when we assume that hyperplane stabilizers are quasi-convex, we do not a priori know that vertex stabilizers are finitely generated (in fact, this is part of the desired conclusion of Theorem A). In both directions, the graph we choose is chosen to reflect the cubical geometry of X, or rather that of X b .
As noted in Remark 2.6 any path in the 1-skeleton of X b has a canonical idealization in X . Each 1-cell e of X b corresponds to some pair of cells ( σ 1 ⊆ σ 2 ) with σ 1 of codimension 1 in σ 2 . If the path p passes over the edge e, its idealization p contains consecutive arrows labelled ( σ 1 ⊆ σ 2 ) → σ 1 and ( σ 1 ⊆ σ 2 ) → σ 2 , and every arrow of p has such a label. Definition 3.6. A pair of opposable scwol arrows in CG(Y) is a pair of scwol arrows γ 1 , γ 2 so that (1) c = i(γ 1 ) = i(γ 2 ) has the property that Θ(c) is a chain ( σ 1 ⊂ σ 2 ) where σ 1 has codimension one in σ 2 ; (2) the label of γ 1 is (1, a) where a is the arrow in Y corresponding to the G-orbit of the arrow ( σ 1 ⊂ σ 2 ) → σ 1 in X ; and (3) the label of γ 2 is (1, b), where b is the arrow in Y corresponding to the G-orbit of the arrow ( σ 1 ⊂ σ 2 ) → σ 2 in X . The center of the pair of opposable arrows (γ 1 , γ 2 ) is the object c = i(γ 1 ) = i(γ 2 ). The image in CG(Y) of a pair of opposable scwol arrows is also referred to as a pair of opposable scwol arrows. Definition 3.7. An object in CG(Y) (equivalently, in Y, since the objects of these two categories are the same) is cubical if it is an orbit of cubes in X (rather than an orbit of chains of cubes of length greater than 1).
The initial and terminal objects of p are cubical; (2) p is a concatenation of group arrows and scwol arrows; and (3) The scwol arrows occur in consecutive pairs, as pairs of opposable scwol arrows. A path in CG(Y) is cubical if its projection to CG(Y) is cubical.
It follows from the definition that all group arrows for a cubical path occur at cubical objects.
The following result is straightforward to prove, starting with an arbitrary CG(Y)path and applying relations until it is of the desired form.
Proposition 3.8. Suppose that v and w are cubical vertices of CG(Y) and σ is a CG(Y)-path between v and w. Then σ is homotopic to a cubical path.
In particular, every g ∈ G = π 1 (CG(Y), v 0 ) is represented by a cubical CG(Y)path starting and ending at v 0 .
These determine a subset S(A) of the arrows of CG(Y) which is the union of the following two sets: (1) S 1 (A) is the set of (group) arrows with label in some A o .
(2) S 2 (A) is the set of scwol arrows occurring in some pair of opposable scwol arrows. As discussed in Section 3.1, there is an associated graph Gr(S(A)) which we denote by Γ(A). A vertex of this graph is called cubical if it comes from a cubical object, and otherwise it is called central.

Note that any central vertex of Γ(A) only meets opposable scwol arrows and thus has valence exactly two, and each of its neighbors is a cubical vertex of Γ(A).
The functor Θ : CG(Y) → X induces a simplicial map All the graphs we construct are subgraphs of Γ(U), and we keep the terminology of cubical vertices and central vertices for these subgraphs.

3.3.
Graphs associated to tuples of intersecting hyperplanes. In this subsection we consider tuples (W 1 , . . . , W k ) of hyperplanes of X b so that i W i is nonempty. We include the possibility of the empty list (), in which case we use the convention that the intersection is all of X b . We observe the following consequence of the cocompactness of G X: Lemma 3.11. There are finitely many G-orbits of finite ordered lists using the convention that the empty intersection of subgroups is G. Definition 3.12 (Choosing representatives). We choose representatives of these ordered lists to preserve inclusion. Namely, first choose a collection of representatives of hyperplanes. If W is a hyperplane, denote its representative by W . Next, choose representatives of pairs (W 1 , W 2 ) with W 1 ∩ W 2 = ∅ so that the representative pair is (W 1 , W 2 ), where W 2 is some hyperplane in the G-orbit of W 2 , though not necessarily W 2 . What we require is that (W 1 , W 2 ) is in the G-orbit of (W 1 , W 2 ), in the sense that there is some g ∈ G so that gW 1 = W 1 and gW 2 = W 2 . More generally, the representative of a list (W 1 , . . . , W k ) starts with the representative of the list (W 1 , . . . , W k−1 ) and then appends an appropriate element of the orbit of W k (though not necessarily the element W k ). Let W denote the chosen finite collection of representative ordered lists of hyperplanes. For 0 ≤ i ≤ dim X, let W i denote the subcollection consisting of ordered lists of length i.
is one of the connected components of the frontier of the carrier of W . As sub-complexes of X b , we write these as the analogous way. It is possible that such a U is a sub-complex of more than one hyperplane, in which case we need to specify which hyperplane we are focusing on. We write U ↑(W ) and U ↓(W ) when there is some ambiguity.
If C is the empty list of hyperplanes, let I C = X b , let V (C) be the set of cubical objects of X b , and let Γ(C) = Γ(U). Now suppose that k ≥ 1. Let C − = (W 1 , . . . , W k−1 ) and suppose by induction that I C − , V (C − ) and Γ(C − ) have been defined. By the way the set W was chosen, C − ∈ W. Define This is obtained by pushing the first hyperplane into a sub-complex of X b , intersecting with the next hyperplane and then pushing this intersection into a sub-complex of X b , and repeating. In particular I C is a subcomplex of I C − . The intersection Define the set The graph Γ(C) is now defined to be Gr (S(C)), where S(C) is the union of the following two sets of arrows: (1) S 1 (C) is the collection of group arrows between objects of V (C).
The following lemmas are clear from the definition.
Lemma 3. 16. For any C ∈ W, Θ(V (C)) lies in the idealization of I C .
The next proposition shows that Γ(C) is Stab(C)-equivariant in a strong sense.
Proof. We continue to use the notation First observe that g preserves Γ(C) if and only if g ∈ Stab(V (C)), defined above in equation ( †).
Suppose that g ∈ Stab(C). Then g ∈ Stab(C − ), so by induction, g preserves V (C − ). To establish that g ∈ Stab(V (C)), it suffices to show that g preserves O(C), the set of idealizations of points in (1) . This holds because gW i = W i for each i ∈ {1, . . . , k}. The first conclusion is proved.
We now assume g Γ(C) ∩ Γ(C) = ∅ and show that g ∈ Stab(C). Suppose that z ∈ g Γ(C) ∩ Γ(C) is a vertex. If z is central, then both its (cubical) neighbors are also in g Γ(C) ∩ Γ(C), so we may assume z is cubical. In The set of labels of group arrows in Ar( Γ(C)) adjacent to v forms a subgroup of G o isomorphic to Stab(C) ∩ Stab (Θ(v)). If v, w are in the same Stab(C)-orbit the corresponding subgroups of G o are equal.
Proof. Fix v ∈ V (C). By the definition of V (C), there is a unique scwol arrow γ with t(γ) = v and i(γ) ∈ Θ −1 (O(C)). Let a be a group arrow from v to w for some w ∈ V (C). There must similarly be a scwol arrow γ with the same label as γ satisfying t(γ ) = w and i(γ ) ∈ Θ −1 (O(C)). In fact we have Θ(γ) = Θ(γ ), so i(γ) is connected to i(γ ) by a group arrow.
Conversely, if v 0 and w 0 are two objects in O(C) which are joined by a group arrow then there is a square with two scwol arrows γ v0 and γ w0 and a group arrow between the corresponding objects in V (C). It is clear that the set of all such group arrows adjacent to v form a group which is isomorphic to G Θ(i(γ)) . This group is isomorphic to Stab(C) ∩ Stab (Θ(v)). The final assertion is clear.  Proof. Suppose C = (W 1 , . . . , W k ), C − = (W 1 , . . . , W k−1 ), as above.
The cubical vertices in Γ(C) are exactly V (C), so we need to show Stab(C) acts cofinitely on V (C).
We claim first that there is a Stab(C)-equivariant bijection α : where Ξ is the set of objects where τ ranges over those cubes of X meeting k i=1 W i . Indeed this can be seen by induction on k. In case k = 0, these sets are equal. Suppose for some cube τ of X. If k = 1, then I C − ∩ τ = τ , and we define α(v) = i(γ 1 ). Otherwise by induction there is a unique scwol arrow γ 2 with t(γ 2 ) = i(γ 1 ) and scwolification For any cube τ meeting The paths without ' ' on them denote the projected CG(Y)-paths.
where p is a cubical CG(Y)-path so that the lift p of p to CG(Y) starting at v C is a concatenation of arrows in Ar( Γ(C)).
Suppose that Stab(C) is generated by a finite set F , and that each g ∈ F is represented by a CG(Y)-path d C · p g · d C as in the conclusion of Lemma 3.21. If each group arrow occurring in each p g has label in Proof. Proposition 3.17 implies that Γ(C) is Stab(C)-invariant. It then follows from the construction that Γ C (B C ) is also Stab(C)-invariant. The free G-action on CG(Y) by deck transformations thus restricts to a free action of Stab(C) on there are only finitely many scwol arrows which begin or end at v. Since B C v is finite, this implies that the graph Γ C (B C ) is locally finite. It now follows from Lemma 3.20 that the Stab(C)-action is cocompact. Now suppose that Stab(C) = F for a finite set F , and that the hypotheses of the final assertion of the proposition is satisfied. We first note that from any vertex v of Γ C (B C ) there is a g ∈ Stab(C) and a path consisting entirely of scwol arrows between v and g. v C . Therefore, it suffices to find a path in Γ C (B C ) between v C an g. v C , for an arbitrary g ∈ Stab(C). We can represent g as a product of elements of F and their inverses. Each of these elements of F is represented by a path based at v 0 of the form d C · p · d C as in Lemma 3.21. Lifting to a path starting at v 0 , these paths determine a path between v 0 and g. v 0 which is homotopic to a path d C · q · d C where q is a cubical path starting at v C which is a concatenation of arrows whose labels all lie in the appropriate B C v , by the choice of F . Therefore, q lies in Γ C (B C ), which proves that Γ C (B C ) is connected, as required.
We denote the restriction of Ψ :

3.4.
If hyperplane stabilizers are QC then cell stabilizers are QC. In this section we prove the direction (1) =⇒ (2) of Theorem A. As mentioned in the introduction, we prove this in greater generality than that of a hyperbolic group acting cocompactly on a CAT(0) cube complex with quasi-convex hyperplane stabilizers. The right general setting for this proof is that of strongly quasi-convex subgroups of finitely generated groups, as defined by Tran in [26]. (Such subgroups were also studied by Genevois [13] under the name Morse subgroups.) Strong quasi-convexity persists under quasi-isometries of pairs. For example, two applications of [26,Proposition 4.2] yields the following result.
Theorem 3.25. Suppose that X and Y are geodesic metric spaces, that A ⊂ X is strongly quasi-convex, that φ : X → Y is a quasi-isometry and that B ⊂ Y lies at finite Hausdorff distance from φ(A). Then B is a strongly quasi-convex subset of Y .
In particular, the notion of strong quasi-convexity makes sense for subgroups of finitely generated groups.
In this subsection, we prove the following theorem.
Theorem 3.26. Suppose that a finitely generated group G acts cocompactly on a CAT(0) cube complex X and that the hyperplane stabilizers are strongly quasiconvex. Then the cell stabilizers are strongly quasi-convex.
Since quasi-convexity is equivalent to strong quasi-convexity for subgroups of hyperbolic groups, Theorem 3.26 immediately implies the direction (1) =⇒ (2) of Theorem A.
Note that each cell stabilizer is a finite intersection of vertex stabilizers. Tran shows that a finite intersection of strongly quasi-convex subgroups is strongly quasiconvex ([26, Theorem 1.2.(2)]) so we only need to show that vertex stabilizers are strongly quasi-convex whenever hyperplane stabilizers are.
We will use the following general statement about intersections of strongly quasiconvex sets, analogous to [8,III.Γ.4.13].
Proposition 3.27. For any Morse gauge M and any D, N, r > 0 there is an R > 0 so that the following holds. Let X be a graph of valence ≤ D with a group G acting on X with at most N orbits of vertices. Let A, B be M -strongly quasi-convex subsets of X satisfying: ( Proof. Note that a concatenation of a geodesic of length r with a geodesic of any length is a (1, 2r)-quasi-geodesic. Let M 0 = M (1, 2r). Let R be the number of pointed oriented simplicial paths in X of length ≤ 2M 0 , up to the G-action.
Let q be the closest point in A ∩ B to p. Suppose d(p, q) > R, and let γ be a geodesic from p to q. Every vertex on γ lies within M 0 of both A and B. By our choice of R, there must be a pair of distinct vertices a 1 , a 2 on γ and paths σ i joining a i to A, and τ i joining a i to B of length at most M 0 , and an element h ∈ G, so that ha 1 = a 2 , hσ 1 = σ 2 and hτ 1 = τ 2 . We may assume that a 1 is closer to q than a 2 is.
Since hA ∩ A and hB ∩ B are nonempty, h must stabilize both A and B. Thus hq ∈ A ∩ B. But hq is closer to p than q is, contradicting our choice of q.
Towards proving Theorem 3.26, suppose that G is a finitely generated group acting cocompactly on a CAT(0) cube complex X, and suppose that hyperplane stabilizers are strongly quasi-convex in G. An index 2 subgroup of a strongly quasiconvex subgroup is strongly quasi-convex, so the stabilizers of hyperplanes in X b are strongly quasi-convex in G. We build an appropriate G-cocompact subgraph of Gr CG(Y) , using the structure of intersections of hyperplane stabilizers. This graph will be Γ C (B C ), for some coherent choices of B C over all of the representative lists C ∈ W (see Definitions 3.12 and 3.22).
Definition 3.28. Suppose that C = (W 1 , W 2 , . . . , W k ) ∈ W is a representative ordered list of hyperplanes as above. An initial segment of C is a list (W 1 , . . . , W i ) for some i < k.
We remark that by the way the representative lists were chosen, if C ∈ W then any initial segment of C is also in W.
Definition 3.29. Suppose that C ∈ W. We define The following holds because the stabilizer of each C is an intersection of hyperplane stabilizers, and because intersections of strongly quasi-convex subgroups are strongly quasi-convex by [  Now choose finite generating sets A C for each Stab(C) ∈ W. By Lemma 3.21, each element g of A C can be represented by a CG(Y)-path d C · p g · d C where p g is a cubical CG(Y)-path so that the lift of p g to CG(Y) starting at v C is a concatenation of arrows in Ar( Γ(C)).
For an object v, choose B C v to consist of the following collections of group arrows: (1) all of the group arrows at v that occur in the paths p g for g ∈ A C ; (2) all of the group arrows at v in the p g for g ∈ A C for any C ∈ I(C); and Use these choices to define a set B C and a graph Γ C (B C ) as in Definition 3.22. The choices made above give us such a graph Γ C (B C ) for each C ∈ W.
Proof. Other than the final statement, the result follows immediately from the construction and Proposition 3.23. The final statement follows immediately from Lemma 3.15 and the second condition in the choice of B C v . Finally, choose a finite generating set A for G, and represent each element of A as a cubical path as in Proposition 3.8. Let A consist of all of the group arrows appearing in these paths, together with all of the group arrows B C for C ∈ W, and use this set to build the graph Γ(A). The following is entirely analogous to Proposition 3.31.
For the remainder of this subsection, we write Γ = Γ(A) and for C ∈ W we write Γ C = Γ C (B C ) for the choices of A and B C as made above.Denote the restriction of the map Ψ : We will need the following lemma in order to apply Proposition 3.27.
This intersection is nonempty since W k intersects I C − nontrivially and Ψ Γ surjects the 1-skeleton of X b .
The group G acts properly and cocompactly on Γ and the strongly quasi-convex subgroup Stab(C) acts properly and cocompactly on Γ C ⊂ Γ. Therefore, by considering an orbit map G → Γ C and applying Theorem 3.25, we see that each Γ C is a strongly quasi-convex subset of Γ. Also, for each hyperplane W of X b the set Ψ −1 Γ (W ) is a strongly quasi-convex subset of Γ. For C ∈ W let M C be a Morse gauge for Γ C , and for W a hyperplane of X b , let M W be a Morse gauge for Ψ −1 Γ (W ). Note that there are finitely many distinct gauges M W as W ranges over all the hyperplanes of X b . Define the Morse gauge M to be the maximum of the M C and the M W .
We now give the main part of the argument of the proof of Theorem 3.26, namely that if hyperplane stabilizers are strongly quasi-convex, vertex stabilizers are also strongly quasi-convex. We therefore fix a vertex v of X.
Note that Ψ −1 Γ (v) is a non-empty and Stab(v)-invariant set of vertices of Γ consisting of finitely many Stab(v)-orbits. Thus in order to show Stab(v) is strongly quasi-convex in G, it suffices (by Theorem 3.25) to show that the pre-image Ψ −1 Γ (v) is a strongly quasi-convex subset of Γ.
We fix constants K ≥ 1 and C ≥ 0, suppose that a and b are vertices in Ψ −1 Γ (v) and let γ be a (K, C)-quasi-geodesic in Γ between a and b. Let y be an arbitrary vertex on γ. We have to show d(y, Ψ −1 Γ (v)) is bounded independent of a and b.
Here is a description of our bound: Let D be a bound for the valence of Γ, N a bound for the number of G-orbits of vertices in Γ. Let R 1 = M (K, C). Assuming R i has been defined, we let R i+1 be the maximum of R i and the constant R in the conclusion of Proposition 3.27, with the above D, N and with r = R i + 1. We will prove that d(y, there is nothing to prove, so we assume that Ψ Γ (y) = v. We will build a sequence of points z 1 , . . . , z t for some t ≤ dim X so that for each i, the following conditions are satisfied: We first find Proof. Since W 1 separates v from Ψ Γ (y), we know that γ must cross Ψ −1 Γ (W 1 ) between a and y. However, Ψ Γ (γ) is a loop, so γ must also cross Ψ −1 Γ (W 1 ) in the segment of γ between y and b. Thus, there is a (quasi-geodesic) subsegment γ 1 of γ which contains y and which starts and finishes on Ψ −1 We claim that z 1 satisfies ( * 1 ) for this choice of g 1 and C 1 . The only thing we have left to prove is that v ∈ g 1 I C1 . This follows immediately from the choice of W 1 .
There are two cases.
We want to apply Proposition 3.27. Take p = y, A = g i+1 Γ Ci and B = Ψ −1 Γ (W i+1 ), and r = R i + 1. The parameters D and N are the maximum valence and number of G-orbits in Γ, respectively.
That gA ∩ A = ∅ implies gA = A is contained in Proposition 3.17.
It is easy to see that if gB∩B = ∅, then gB = B, since a hyperplane is determined by any edge dual to it. Lemma 3.33 asserts that A ∩ B = ∅. By the definition of R i+1 , Proposition 3.27 implies , so we have established ( * i+1 ) and finished the proof.
For j > dim X, there cannot exist a point z j satisfying ( * j ), since there are no j-tuples of hyperplanes with nonempty intersection. Therefore, Proposition 3.35 asserts that for some This completes the proof of Theorem 3.26.
3.5. If cell stabilizers are QC then hyperplane stabilizers are QC. In this section we prove the direction (2) =⇒ (1) of Theorem A. Therefore, suppose that G is a hyperbolic group acting cocompactly on a CAT(0) cube complex X, and suppose that the vertex stabilizers are quasi-convex in G. In particular, these vertex stabilizers will be finitely generated. Note that stabilizers of other cells are intersections of vertex stabilizers, so they are also quasi-convex, and so finitely generated.
Let W be a hyperplane in X b . We simplify the notation set up in Section 3.3 by writing 'W ' instead of the 1-tuple '(W )'. Lemma 3.36. Let v ∈ V (W ), and let H v,W be as in Definition 3.19. There is a cube σ of X so that H v,W is naturally isomorphic to a finite-index subgroup of G σ .
Proof. The stabilizer in G of some v ∈ V (W ) naturally corresponds to the stabilizer of a pair (σ 0 ⊂ σ), a codimension-1 inclusion of cubes of X. But Stab(W ) ∩ Stab (Θ W (v)) is exactly the stabilizer of (σ 0 ⊂ σ), which has finite-index in the stabilizer of σ, which in turn is isomorphic to G σ .
Form the graphs Γ W = Γ W (B W ) and Γ = Γ(A) as described in Definitions 3.9 and 3.22.
Proposition 3.37. The graph Γ is connected and G-invariant and G acts freely and cocompactly on Γ. The graph Γ W is connected and Stab(W )-invariant, and Stab(W ) acts freely and cocompactly on Γ W . Moreover, Γ W ⊂ Γ.
Proof. The special cases that C = () and C = (W ) of Proposition 3.23 give the first two sentences of the proposition. The final assertion follows quickly from the condition that Given a vertex w of W ↓ , let Y (w) be the (closed) 1-neighborhood in Γ W of Ψ −1 W (w). Lemma 3.38. The sets Y (w) are quasi-convex subsets of Γ with constants which do not depend on w.
Proof. Since Stab(W ) acts cocompactly on W ↓ , there are finitely many Stab(W )orbits of sets Y (w), so the uniformity of constants will follow immediately if we can prove each Y (w) is a quasi-convex subset of Γ.
The stabilizer in Stab(W ) of Y (w) is the same as the subgroup H w,W from Definition 3.19. By Lemma 3.36 this is a finite index subgroup of some cell group of the G-action on X. Thus, by hypothesis, H w,W is a quasi-convex subgroup of G. Since G acts properly and cocompactly on Γ, and H w,W acts properly and cocompactly on Y (w) ⊂ Γ, the result follows (for example, by Theorem 3.25).
We are ready to prove the direction (2) =⇒ (1) of Theorem A, which is the content of the following theorem. For this result, we assume Theorem A.3, which is proved in Appendix A.
Theorem 3.39. Suppose that the hyperbolic group G acts cocompactly on the cube complex X, and that for every vertex v of X, the stabilizer Stab(v) is quasi-convex. Then, for every hyperplane W ⊂ X, the stabilizer Stab(W ) is a quasi-convex subgroup of G.
Proof. As we have already remarked, quasi-convexity of vertex stabilizers implies quasi-convexity of all cell stabilizers.
Let Γ, Γ W and the Y (w) be as discussed above. Since G acts freely and cocompactly on Γ, we know that Γ is δ-hyperbolic for some δ. Let be a constant so that Y (w) is -quasi-convex for every w (Lemma 3.38).
Since Stab(W ) acts freely and cocompactly on Γ W , in order to prove the theorem it suffices to prove that Γ W is quasi-convex in Γ, so let p, q ∈ Γ W .
Consider a geodesic γ in (X b ) (1) between Ψ(p) and Ψ(q). Both Ψ(p) and Ψ(q) lie in W ↓ . Since W ↓ is combinatorially convex in X b , the geodesic γ is entirely contained in the 1-skeleton of W ↓ . The vertices w 1 , . . . , w n on γ correspond to cells of X contained in W ↓ . The sets Y (w i ) corresponding to these cells satisfy the hypotheses of Theorem A.3 with m = 2, c = 1, and the quasi-convexity constant chosen above. Theorem A.3 then implies that Y (w 1 ) ∪ · · · ∪ Y (w n ) is -quasi-convex, for a constant depending only on and δ.
In particular, a Γ-geodesic between p and q lies within of Y (w 1 ) ∪ · · · ∪ Y (w n ). Since each of these Y (w i ) is contained in Γ W , the Γ-geodesic between p and q stays uniformly close to Γ W , as required.
Together with Theorem 3.26, this completes the proof of Theorem A.
3.6. On generalizations of Theorem A. For a subgroup H of a hyperbolic group G, the following three conditions are equivalent: (a) H is strongly quasi-convex in G.
(c) H is undistorted in G. Dropping the condition that G is hyperbolic, condition (b) ceases to be well-defined, but the conditions (a) and (c) still make sense.
One can ask for versions of Theorem A where the hypothesis of hyperbolicity is removed and condition (b) is replaced by either condition (a) or (c).
3.6.1. Strong quasi-convexity. Replacing quasi-convexity with strong quasi-convexity we can ask about the following conditions for a finitely generated group G acting cocompactly on a CAT(0) cube complex: (1S) Hyperplane stabilizers are strongly quasi-convex. The remaining implication (3S) =⇒ (1S) is false, as the example of Z 2 acting freely on a cubulated R 2 shows.
3.6.2. Undistortedness. The situation when replacing quasi-convexity with quasiisometric embeddedness is murkier. We consider the following conditions, for a finitely generated group G acting cocompactly on a CAT(0) cube complex X: (1U) Hyperplane stabilizers are undistorted.
(3U) All cell stabilizers are undistorted. If X is a tree, (1U) and (3U) each implies (2U), but not conversely. For example, the double of a finitely generated group over a distorted group acts on a tree with undistorted vertex stabilizers but distorted edge/hyperplane stabilizers.
We do not know the relationship between (1U) and (3U) in general, so we ask the question. Definition 3.41. [Height of a family] Suppose that G is a group and H is a collection of subgroups. The height of H is the minimum number n so that for every tuple of distinct cosets (g 0 H 0 , g 1 H 1 , . . . , g n H n ) with H i ∈ H (and g i ∈ G), the intersection ∩ n i=0 H gi i is finite. If there is no such n then we say the height of H is infinite.
In case H = {H} is a single subgroup, we recover the familiar notion of the height of a subgroup from [14].
The following result for a single subgroup is part of [14,Main Theorem]. The proof of that result from [1] (Corollary A.40 in that paper) can be adapted in the obvious way to prove the result for finite families. This result was proved in the more general setting of strongly quasi-convex subgroups by Tran [  We also use the following special case of a theorem of Charney-Crisp [9, Theorem 5.1].
Theorem 3.43. Suppose that G acts cocompactly on a cube complex X. Then X is quasi-isometric to the space obtained from the Cayley graph of G by coning cosets of stabilizers of vertices to points.
We now prove Corollary B. For convenience, we recall the statement.
Corollary B. Suppose that G is a hyperbolic group acting cocompactly on a CAT(0) cube complex X with quasi-convex hyperplane stabilizers. Then (1) X is δ-hyperbolic for some δ; (2) there exists a k ≥ 0 so that the fixed point set of any infinite subgroup of G intersects at most k distinct cells; and (3) the action of G on X is acylindrical (in the sense of Bowditch [6, p. 284]).
Proof. If G is a hyperbolic group acting cocompactly on a CAT(0) cube complex, and if the stabilizers in G of vertices in X are quasi-convex, then [7, Theorem 7.11], due to Bowditch, implies that this coned graph is δ-hyperbolic for some δ. Theorem 3.43 then implies that the cube complex X is δ-hyperbolic for some (possibly different) δ. Thus, we have the first statement from Corollary B. Now we prove the statement about fixed point sets of infinite subgroups. Let I be a collection of orbit representatives of cells in X. For i ∈ I, let Q i = {g ∈ G | gi = i}, and let Q = {Q i } i∈I . Then Q is a finite collection of quasi-convex subgroups of G, so it has some finite height k by Proposition 3.42. If H < G is infinite with nonempty fixed set and σ is a cell meeting the fixed point set of H, then H < Q g i where σ = gi. Since the height of Q is k, at most k such cells appear. In [12], Genevois studies actions of groups on hyperbolic CAT(0) cube complexes and shows in Theorem 8.33 that, in this setting, acylindricity is equivalent to the condition: We take R to be the maximum size of a finite subgroup and L = k. Suppose d(x, y) ≥ L. Then the union of the combinatorial geodesics joining x to y contains finitely many (but at least k + 1) vertices. There is a finite index subgroup of Stab(x) ∩ Stab(y) which fixes all of these vertices. This finite index subgroup fixes more than k cells, so it is finite. This implies Stab(x) ∩ Stab(y) is finite, as desired.
Remark 3.44. In the context where G is a finitely generated group acting cocompactly on a cube complex X with strongly quasi-convex hyperplane stabilizers, the same proof of conclusion (2)

Conditions for quotients to be CAT(0)
As noted in the introduction, Theorem D follows quickly from Theorem A, Theorem F, Agol's Theorem [1, Theorem 1.1] and Wise's Quasi-convex Hierarchy Theorem [27,Theorem 13.3]. Thus, other than Theorem A.3 in Appendix A (which is independent of everything else in this paper), it remains to prove Theorem F. Therefore, we are interested in conditions on a group G acting on a CAT(0) cube complex X and a normal subgroup K G which ensure that the quotient K X is a CAT(0) cube complex. In this section we develop criteria in terms of complexes of groups to ensure this. In the next section, we translate these conditions into algebraic conditions on K G.
Three conditions need to be ensured in order for the complex X = K X to be a CAT(0) cube complex: (1) X must be simply-connected; (2) X must be a cube complex (rather than a complex made out of cells which are quotients of cubes); and (3) X must be non-positively curved. We investigate these three properties in turn.

4.1.
Ensuring the quotient is simply-connected. First, we give a sufficient condition for K X to be simply-connected.
Since X is a finite dimensional cube complex, it has finitely many shapes, and we can use the following application of a theorem of Armstrong: Theorem 4.1. Let X be a simply connected metric polyhedral complex with finitely many shapes, and let K be a group of isometries of X respecting the polyhedral structure, generated by elements with fixed points. Then K X is simply connected.
Proof. (Sketch) A theorem of Armstrong, [4, Theorem 3], shows that K X is simply connected with the CW topology. We have to show it is still simply connected with the metric topology.
Because X has finitely many shapes there is an equivariant triangulation T and an > 0 so that for every finite subcomplex K, the -neighborhood of K deformation retracts to K. If f : S 1 → X is any loop, then a compactness argument shows it lies in an -neighborhood of some such finite complex. We can then homotope f to have image in K and apply the simple connectedness of K X with the CW topology.
We remark that the hypothesis of finitely many shapes is necessary even when X is CAT(0) as the following example shows: Example 4.2. For n ∈ {2, 3, . . .}, let D n be the Euclidean cone of radius 1 on a loop σ n of length 2π n 2 . For each n mark a point on σ n . Let Y be obtained from D n by identifying the marked points. Unwrapping all the cones to Euclidean discs gives a tree of Euclidean discs of radius 1. We call this CAT(0) space Y . There is a discrete group of isometries Γ = γ 2 , γ 3 , . . . acting on Y with quotient Y , so that each γ n fixes the center of some disc and rotates it by an angle of 2π n 2 . Nonetheless Y is not simply connected, as the infinite concatenation of the loops σ n has finite length, but cannot be contracted to a point.

4.2.
Ensuring the quotient is a cube complex. We now turn to the question of when K X is a cube complex.
In order that the quotient Z = K X be a cube complex, there needs to be no element of K which fixes a cell of X set-wise but not point-wise.
Suppose that σ is a cube of X. The stabilizer G σ has a finite-index subgroup Q σ consisting of those elements which fix σ pointwise. Let {σ 1 , . . . , σ k } be a set of representatives of G-orbits of cubes in X. The following result is straightforward. Proposition 4.3. Suppose that G acts cocompactly on the cube complex X and that K is a normal subgroup of G so that for each i we have G σi ∩ K ≤ Q σi . Then the quotient K X is a cube complex and the links of vertices in K X inherit a cellular structure from the simplicial structure of cells in X.

4.3.
Ensuring the quotient is nonpositively curved. The most complicated condition to ensure is that K X is nonpositively curved.
Throughout this subsection we suppose that X is a CAT(0) cube complex and that X is its idealization (see Definition 2.2). We suppose further that G is a group acting cocompactly on this cube complex. The induced action of G on X has quotient a scwol Y. Making choices as in Definition 2.8, we obtain a complex of groups G(Y), with associated category CG(Y). Choosing a vertex v 0 ∈ Y, there is then an identification of G with π 1 (CG(Y), v 0 ). Moreover, we choose a normal subgroup K G so that K X is a cube complex. 3 In Subsection 4.2 we discuss how to find subgroups K so that K X is a cube complex, but for this section we just assume that this is the case.
Let C K be the cover of the category CG(Y) corresponding to the subgroup K. Observe that CG(Y)-loops lift to C K if and only if they represent elements of K. (Basepoints are mostly omitted in this section, since we deal with a normal subgroup K.) Standing Assumption 4.4. Through this section we write CG(Y)-paths as a concatenation of group arrows and scwol arrows (which need not alternate between group arrows and scwol arrows). Thus in a list of arrows such as g 1 ·e 1 ·e 2 ·g 2 ·..., each g i is an element of a local group G v and represents the edge (g, 1 v ) + , corresponding to the group arrow (g, 1 v ). The e i represent a ± i for a scwol arrow (a i , 1), and we blur the distinction between the scwol arrow (a i , 1) in CG(Y) and the Y-arrow a i , and also between the CG(Y)-path (1, a i ) ± and the Y-edge a ± i . We implicitly assume that each concatenation we write defines a path, which often forces the group arrows labelled g i to be elements of particular local groups. Whenever we consider a CG(Y)-path of length 1 consisting of a single group arrow we are either explicit about the local group or else it is clear from the context.
In case we have a group arrow of the form (1, 1 v ), we often implicitly (or explicitly) omit this arrow from our path. In this section, we provide a set of conditions on the subgroup K which imply the link condition for Z.
The link of a cube σ in a cube complex has a natural cellulation by spherical simplices, one coming from each inclusion of σ into a higher-dimensional cube. In particular lk(σ) is a ∆-complex [19, Chapter 2.1] (though it may not be simplicial).
We record two elementary observations: Proof. If L fails to be simplicial, there is either a non-embedded simplex, or a pair of simplices which intersect in a set which is not a face of both. If a simplex is non-embedded, we obtain a loop of length 1 in L. If two embedded simplices τ 1 and τ 2 of L intersect in a set which is not a single face, let F 1 and F 2 be different maximal faces in the intersection, and let f = F 1 ∩ F 2 . For i ∈ {1, 2}, let v i be a vertex in F i \ f . Then the simplices spanned by v 1 ∪ f and v 2 ∪ f correspond to points in lk(f ) which lie on an immersed loop of length 2. But f corresponds to some cube containing σ, and lk(f ) ⊂ L is isomorphic to the link of that cube. Therefore, in order to ensure Z is nonpositively curved, for each cell σ in Z we must rule out loops of length 1 and 2 in lk(σ) and also ensure that any loop of length 3 in lk(σ) is filled by a 2-cell. We first explain how we translate between 1-cells in links in Z and CG(Y)-paths. Then we develop the required conditions to rule out loops, finally dealing with loops of length 3 which must be filled by 2-cells.

CG(Y)-paths associated to 1-cells in lk(σ)
. Below we choose, for each cube σ in Z and each 1-cell α in the link of σ, a CG(Y)-path p α which is the label of an unscwolification of the idealization of α. As indicated by the notation, this label is the same for two such 1-cells in the same G-orbit. (In fact we will choose these paths for slightly more general objects than 1-cells in links of cubes.) First fix a cube σ of Z. The second barycentric subdivision of the link of σ embeds naturally in the geometric realization of Z. The vertices of the image of lk(σ) are precisely the length ≥ 2 chains of cubes whose minimal element is σ.
In particular, an oriented 1-cell α of lk(σ) has idealization a Z-path of length 4, made up of arrows In what follows, we want a slightly more general situation, where 1 , 2 , φ are cubes in Z with 1 , 2 codimension-1 sub-cubes of φ, and γ is a chain of cubes in Z so that each element of γ is contained in each of 1 , 2 , φ. We can naturally extend γ to chains which we denote (γ ⊂ 1 ), (γ ⊂ 2 ), (γ ⊂ φ), and (γ ⊂ i ⊂ φ), and this triple of cubes correspond to a 1-cell in an 'iterated link' (a link of a cell in a link, etc.), and also has idealization a Z-path of length 4 as follows: The Z-path (2) may not embed in Y. There are two ways this could happen. The first is that there is an element of Stab(γ) which sends 1 to 2 , but no such element fixes φ. In this case, the image in Y is a non-backtracking loop. The second possibility is that there is an element g ∈ G sending each of γ and φ to itself, but exchanging 1 and 2 . If there is such a g, the idealization of the 1-cell α backtracks in Y, forming a 'half-edge'.
Let y α = a + 1 ·a − 2 ·a + 3 ·a − 4 be the Y-path which is the image of the Z-path above. Let ν be the projection of (γ ⊂ φ) in Y, and µ i be the projection of (γ ⊂ i ⊂ φ). Let ξ i be the projections of (γ ⊂ i ). Then we have the injective homomorphisms The images of ψ a2 and ψ a3 are equal. The projections of all the data associated to α depend only on the orbit of α under the stabilizer of σ. We shall denote this orbit by α , and denote the common image of ψ a2 and ψ a3 in G ν by G + ν . Note that G + ν either has index 2 in G ν (in case there is a g fixing φ and exchanging 1 with 2 ) or else G + ν = G ν (if there is no such g). In case G + ν has index 2 in G ν , we fix a choice of g ν ∈ G ν G + ν . We make this choice once and for all for each orbit of (γ, 1 , 2 , φ), so that the choice depends only on the orbit and not on the representative.
In the sequel, we refer to the vertex groups by G i( α ) (for G ξ1 ) and G t( α ) (for G ξ2 ). We further define "edge-inclusions" ψ α : G + ν → G t( α ) and ψ α : ν is the element fixed above. Note that in this case a 2 = a 3 and a 1 = a 4 . In case G + ν = G ν let p α = a + 1 · a − 2 · a + 3 · a − 4 . In either case some lift of p α to C K is an unscwolification of the idealization of α.
In both cases, the scwolification of the path is fixed during the homotopy. Moreover any lift of the homotopy to a cover of CG(Y) gives a sequence of paths with constant scwolification. Remark 4.10. We remark that in case G + ν = G ν the paths considered in the second half of the above statement are exactly those CG(Y)-paths traversing y α which lift and scwolify (using Θ) to non-backtracking paths in X (see the discussion at the end of Section 2.5). As Z = K X is a cube complex, these CG(Y)-paths also lift and scwolify (using Θ K ) to non-backtracking paths in Z.
Notation 4.11. We fix some notation in order to study paths in Z and also Ypaths and CG(Y)-paths. As above, we use . to denote a G-orbit in Z, which corresponds to its image in Y under the projection π : Z → Y.
Let p α be one of the CG(Y)-paths fixed in Definition 4.8, corresponding to a 1-cell α in some link (or iterated link) of a cube of Z. The CG(Y)-path p α has an underlying Y-path, which we denote by y α . Define t( α ) = t(y α ) and i( α ) = i(y α ). This is so we can denote the corresponding local groups as G i( α ) and G t( α ) .
We will also need to refer to the subgroups E α < G t( α ) and E α < G i( α ) defined just before Definition 4.8. Each of these subgroups can be thought of as the pointwise stabilizer of some translate of a lift of α to X.

4.3.2.
Loops in lk(σ). We are now ready to formulate the conditions on K which characterize whether or not K X is non-positively curved. We use Lemmas 4.6 and 4.7 repeatedly.
Recall that we have fixed a K G so that Z = K X is a cube complex. We also fix a cube σ of Z. If α is a 1-cell in lk(σ), there is a corresponding Z-path of length 4, and we sometimes conflate the two.
We next give an algebraic characterization of loops of length 1 in lk(σ). Proof. Thinking of Z as the geometric realization of Z, the 1-cell α is the realization of a Z-path q α of length 4, which projects to a Y-path a + 1 · a − 2 · a + 3 · a − 4 . Let q α be an unscwolification of q α in C K , which we may choose to have label (3) a + 1 · a − 2 · g 1 · a + 3 · a − 4 , for some group arrow g 1 .
Suppose first that the endpoints of α coincide. Then the path (3) has endpoints separated by a group arrow, and so there is a C K -loop with label a + 1 · a − 2 · g 1 · a + 3 · a − 4 · g 2 . Lemma 4.9 implies that this loop is homotopic to a loop with label p α · g for some g.
Conversely, suppose a conjugacy class in K is represented by a CG(Y)-loop of the form p α ·g. Then p α ·g lifts to a loop in C K whose scwolification is a translate of q α by some element of G. In particular, q α must be a loop, and so the endpoints of α coincide.
Definition 4.14. A CG(Y)-path p is K-non-backtracking if for some (equivalently any) lift p to C K , the scwolification Θ K ( p) is non-backtracking. A CG(Y)-loop can be thought of as a path starting at any of its vertices. If all these paths are Knon-backtracking, we say that the loop is K-non-backtracking. Lemma 4.15. A CG(Y)-path g 0 · p α1 · g 1 · p α2 · . . . · g k−1 · p α k · g k is Knon-backtracking if and only if the first of the following two conditions holds. A CG(Y)-loop with such a label is K-non-backtracking if and only if both conditions hold. ( The following result algebraically characterizes immersed loops of length 2 in lk(σ). (1) There is a path p = α .β in lk(σ) with α = α and β = β so that p is an immersed loop. (2) There is a K-non-backtracking CG(Y)-loop p α ·g 1 ·p β ·g 2 that represents a conjugacy class in K. Moreover, in case these conditions hold, the path p can be chosen to be the scwolification of a lift of p α · g 1 · p β · g 2 (and conversely p α · g 1 · p β · g 2 is the CG(Y)-path which labels the unscwolification of p ).

Proof. Suppose that there is a immersed loop
as discussed above. Using Lemma 4.9, we can choose an unscwolification q p of q p in C Y with label p α · g 1 · p β , where g 1 is a group arrow. But the unscwolification q p has endpoints separated by a group arrow g 2 , so there is a loop labeled p α · g 1 · p β · g 2 as desired. It is K-non-backtracking since its scwolification is the path q p .
Conversely, suppose that there is a K-non-backtracking CG(Y)-loop which represents an element of K. Then p α · g 1 · p β · g 2 lifts to a loop in C K . The scwolification of this loop gives a path p as in condition (1).
The following is elementary. The utility of Lemma 4.17 is that once we have found conditions to ensure that links in K X have no edge-loops of length 1 or 2 then edge-loops of length 3 are automatically non-backtracking.
Given Lemma 4.17, the following is proved in the same way as Lemma 4.16.
Lemma 4.18. Suppose that lk(σ) is simplicial, and suppose that p is a path in lk(σ) which is a concatenation of three 1-cells, α, β and γ. The following are equivalent: (1) There is a path p = α .β .γ in lk(σ) so that α = α , β = β and γ = γ , and p is an immersed loop. (2) There is a CG(Y)-loop of the form p α · g 1 · p β · g 2 · p γ · g 3 that represents a conjugacy class in K.
Moreover, in case these conditions hold, the path p can be chosen to be the scwolification of a lift of p α · g 1 · p β · g 2 · p γ · g 3 (and conversely p α · g 1 · p β · g 2 · p γ · g 3 is the CG(Y)-path which labels the unscwolification of p ).
If X has dimension greater than 2, there are certainly some σ so that there are loops of length 3 in lk(σ). This introduces some subtleties, which we discuss in the next subsection.  Let Y be a single 2-simplex, and consider the complex of groups G(Y) so that G v ∼ = Z for each vertex v, and all the other local groups are trivial. Let x, y, z ∈ π 1 (G(Y)) generate the three vertex groups. The universal cover X of G(Y) is an infinite valence "tree of triangles". Let K = x 3 , y 3 , z 3 , xyz . Then K X can be realized as a subset of the Euclidean plane, consisting of every other triangle of a tessellation by equilateral triangles. Moreover, if αβγ is the path in the 1-skeleton of Y labeling the boundary of Y, there are paths in K X projecting to αβγ, but which are not filled by a 2-cell in K X . The issue here, as we will see, is that xyz ∈ K is not an element of K x K y K z , where K x = K ∩ x , and so on.
Of course X is not a cube complex, but it can be realized as the link of a vertex of a cube complex, covering a complex of groups in which G(Y) is embedded.
Definition 4.20. Let σ be a cube of Z, and let τ be a 2-cell in lk(σ). Then ∂τ is a loop composed of three oriented 1-cells α.β.γ. These 1-cells are associated to CG(Y)-paths p α , p β , p γ as in Definition 4.8. Consider a CG(Y)-path of the form q = p α · g · p β . Let q be a lift to C K . The realization of Θ K ( q) is a concatenation of two 1-cells α .β . We say that q K-bounds a (τ, α)-corner if there is a cube σ , a 2-cell τ in lk(σ ), and an h ∈ G so that σ = hσ, τ = hτ , α = hα and β = hβ. If there is some (τ, α) for which the path q K-bounds a (τ, α)-corner, we may just say q K-bounds a corner.
(which is based at σ ⊂ ) represents an element of G σ⊂ .
Proof. All the chains which occur in this proof have the same minimal element σ, so we omit the prefix 'σ ⊂' from all chains until the end of the proof of the Lemma. We therefore have a diagram in link(σ) in the scwol Z as follows: a α a β Figure 1. A part of Z representing part of the link of σ, containing the idealization of the 1-cell ζ in green. Directions of most arrows have been omitted.
( ) We have the following identities of morphisms in the category Z: a α a 1 = b 1 = b 2 a 2 and a β a 4 = b 3 = b 2 a 3 . The path in the statement of the lemma is equal to: where g ( ⊂ψ) is the element of G ( ⊂ψ) chosen for the path p ζ as in Definition 4.8. Define the following elements of G : where the z( a , b ) are the twisting elements determined by the complex of groups structure on G(Y).
We now have the following sequence of elementary homotopies of CG(Y)-paths (all of which consist of applying the moves in Definition 2.4, and the rule of arrow composition in CG(Y) from Definition 2.12).
This proves the result.
Notation 4.22. The element of G represented by a α + · p ζ · a β − is denoted by g τ,α . Lemma 4.23. A path p α · g · p β K-bounds a (τ, α)-corner if and only if there exists a CG(Y)-loop which represents an element of K.
Proof. First suppose that there is a CG(Y)-loop of the form (4) representing an element of K. Then the following two CG(Y)-paths differ by an element of K: and p α · a α + · g 1 · p ζ · g 2 · a β − · g −1 · g · p β .
Thus they together form a loop which lifts to C K . This second path is homotopic to a CG(Y)-path whose scwolification avoids the vertex t( α ) after p α but instead travels across the first three edges of p α , traverses p ζ , and then travels across the final three edges of p β . The homotopy lifts to C K , and the image in Z of this homotopy under the scwolification Θ K shows that there is a 2-cell τ between the edges α and β which are the images of the lifts of p α and p β respectively. This shows that the path p α · g · p β K-bounds a (τ, α)-corner. Now suppose that the CG(Y)-path q = p α · g · p β K-bounds a (τ, α)-corner. Lift to a C K -path q and consider the scwolification Θ K ( q) in Z. As in Definition 4.20, the realization of q is the concatenation of two 1-cells α , β in lk(σ ) for some cube σ in the orbit of σ. Moreover, there is a 2-cell τ with α , β in the boundary of τ and an element h of G so that σ = hσ, τ = hτ , α = hα and β = hβ. Let v be the vertex of lk(σ ) where α and β meet.
Consider the loop q 0 = a α + · p ζ · a β − as in Lemma 4.21. This represents an element of G t( α ) , and there is a lift q 0 of q 0 to C K so that q 0 = Θ K ( q 0 ) is a loop based at v and traveling across the corner of τ from α to β . The paths q 0 and q have lifts to C K forming a sub-diagram: ? e e --q q 9 9 ? ? _ _ The circled dots represent either single objects or pairs of objects separated by a group arrow, depending on whether the paths p x have length four or five for x ∈ {α, β, ζ}. The scwolification of this diagram in Z looks like this: The edges which scwolify to a α in q and q 0 have sources connected by a group arrow labeled by some g 1 . Similarly the edges which scwolify to a β have sources connected by group arrow with some label g 2 . We thus obtain a loop in C K of the form (4).
Given the criterion from Lemma 4.23, the following result is straightforward. Recall the definition of the element g τ,α from Notation 4.22.
Proposition 4.24. Suppose that τ is a 2-cell in lk(σ) and that the boundary of τ is α.β.γ. For any g ∈ G t α the CG(Y)-path p α ·g ·p β K-bounds a (τ, α)-corner if and only if Proof. Recall from Notation 4.22 that g τ,α is the element of G t α represented by the CG(Y)-loop a α + · p γ · a β − .
We have homotopies Since ψ aα (g 1 ) ∈ E α , ψ a β (g 2 ) ∈ E β and the whole expression above is an as required.
In order to prove the other direction, this computation may be performed in reverse.
Lemma 4.25. Suppose that lk(σ) is simplicial and contains 1-cells α, β, and γ. Let q = p α · g 1 · p β · g 2 · p γ · g 3 be a CG(Y)-loop which represents an element of K. Suppose q ⊂ lk(σ) is the realization of the scwolification of some lift of q to C K .
Proof. Note that since q represents an element of K, any lift to C K is a loop, and so the realization q is also a loop. Since lk(σ) is simplicial, this loop is embedded of length 3 in lk(σ) by Lemma 4.17.
Think of q as given by a cyclic word in the arrows of CG(Y), and suppose that one of the three given subpaths of q K-bounds a corner. By relabelling and cyclically rotating we can assume it is the subpath p = p α · g 1 · p β , so there is some 2-cell τ and p K-bounds a (τ, α)-corner. It follows that some translate τ of τ in lk(σ) has boundary given by a path α .β .γ , where α .β are the first two 1-cells of the path q. If the third 1-cell of ∂τ is not the third 1-cell of q, we obtain 1-cells in lk(σ) with the same endpoints, contradicting the assumption that lk(σ) is simplicial. So q bounds the 2-cell τ .
Since there are finitely many Stab(σ)-orbits of 2-cell in lk(σ), we obtain the following.
which represents an element of K, there exists an i so that (1) α = α i , β = β i and γ = γ i ; (2) g 1 ∈ E αi g τi,αi E βi K t( αi ) ; (3) g 2 ∈ E βi g τi,βi E γi K t( βi ) ; and (4) g 3 ∈ E γi g τi,γi E ai K t( γi ) ; Proof. Choose the 2-cells τ i to be representatives of the Stab(σ)-orbits of 2-cells (together with a fixed vertex to label the boundary -so that a single orbit may appear up to three times in the list).
Suppose first that the condition about paths of form ( * ) representing elements of K is satisfied, and suppose that p is an edge-loop of length 2 in lk(σ) which is labelled by 1-cells α , β , γ , in order. By Lemma 4.18 there exists a CG(Y)path λ of the form ( * ) which is the label of an unscwolification of p. Because of our hypothesis, there exists an i so that conditions (1)-(4) are satisfied. By Proposition 4.24 the CG(Y)-path λ K-bounds a corner at each of its three corners, and so by Lemma 4.25 the path p bounds a 2-cell, as required.
Conversely, suppose that every edge-loop of length 3 in lk(σ) bounds a 2-cell, and consider a CG(Y)-path λ of the form ( * ) which represents an element of K. By Lemma 4.18 the scwolification p of λ is an immersed edge-path of length 3, which hence must bound a 2-cell, τ say. Suppose that τ i is the representative in the Stab(σ)-orbit of the 2-cell τ , so condition (1) is satisfied. According to Lemma 4.23, applied to all three corners of this 2-cell, the path λ satisfies conditions (2)-(4). This finishes the proof.
To summarize, Lemmas 4.12, 4.16, 4.18, and Proposition 4.26 give descriptions of various types of CG(Y)-paths so that the cube complex Z = K X is nonpositively curved if and only if no such path lifts to C K .

Algebraic translation
In this section, we continue to work in the context of a group G acting cocompactly on a CAT(0) cube complex X. The induced action on the associated scwol X has quotient scwol Y, the underlying scwol for a complex of groups structure G(Y) on G. We let Q(G) be the set of cube stabilizers for G X; equivalently Q(G) is the set of conjugates of the local groups for the complex of groups G(Y).
We translate the conditions from the previous section into algebraic statements about elements of G and of Q(G), with an eye toward finding conditions on K G so that K X is non-positively curved. In Section 6 we use hyperbolic Dehn filling to find K which satisfy the conditions, under certain hyperbolicity assumptions on G and Q(G).
We fix a basepoint v 0 for Y and an isomorphism π 1 (CG(Y), v 0 ) ∼ = G as in Section 2. The scwolification functor Θ : CG(Y) → X is G-equivariant. Recall also that the objects of CG(Y) are homotopy classes of paths starting at v 0 .
Fix also a maximal (undirected) tree T in Y. For each object v of Y which represents an orbit of cubes in X, let c v be the unique Y-path in T from v 0 to v. By using scwol arrows, we consider c v also to be a CG(Y)-path in the natural way. For an object v of Y which represents a chain of cubes of length longer than 1, we define a Y-path c v from v 0 to v as follows: If v is represented by (σ 1 ⊂ σ 2 ⊂ · · · ⊂ σ k ) (a nested chain of cubes in X) then define c v to be the concatenation of c σ1 with the path consisting of the arrows (σ 1 ⊂ · · · ⊂ σ i ) → (σ 1 ⊂ · · · ⊂ σ i+1 ), for i = 1, 2, . . . , k − 1.
We use the paths c v to define a map from (homotopy classes rel endpoints of) CG(Y)-paths to (homotopy classes of) CG(Y)-loops based at v 0 by Given a path p, let p = c i(p) · p · c t(p) ∈ π 1 (CG(Y), v 0 ).
The following results are all straightforward.

5.1.
Algebraic formulation of the link conditions. Suppose that K G. In order for Z = K X to be non-positively curved, there are five conditions that need to be ensured on links in Z. Roughly speaking, they are: More precisely, the "image in Y" means the image in Y of the idealization. And we say this image p "bounds a 2-cell" if there is an unscwolification p and a lift p of p to CG(Y) so that the realization of the scwolification of p bounds a 2-cell in some link of a cube in X.
If K X is a simply-connected cube complex and we ensure each of these conditions, then Lemmas 4.6, 4.7 and 4.17 imply that K X is CAT(0).
In this subsection, we formulate five results which give algebraic conditions to enforce each of these five conditions in turn. These results follow quickly from the results in Section 4 using the translation from the beginning of this section. In each case, since G acts cocompactly on a CAT(0) cube complex, there are finitely many G-orbits of links and in each link finitely many G-orbits of each of the five kinds of paths in the above list, and we can rule out each orbit behaving badly in K X in turn.
Assumption 5.4. The group G acts cocompactly on the CAT(0) cube complex X, and Q(G) is the collection of cell stabilizers of the action.
Terminology 5.5. Under Assumption 5.4, a normal subgroup K G is co-cubical if K X is a cube complex.
The following is a straightforward translation of Lemma 4.12. We spell out the proof since we use similar techniques for other more complicated results later in the section.
Theorem 5.6. Under Assumption 5.4 there exists a finite set F 1 ⊂ Q(G) × G so that for each (Q, p) ∈ F 1 we have p ∈ Q and so that if (i) K G is co-cubical; and (ii) for each (Q, p) ∈ F 1 we have p ∈ Q.K, then no link in K X contains an edge-loop of length 1.
Proof. Up to the action of G, there are finitely many pairs ( σ, α), where σ is a cube of X and α is a 1-cell in link( σ) whose endpoints are identified by some element of G. For each such pair we will give a pair (Q, p) as in the statement of the theorem.
For such a pair, let (σ, α) be the image in K X . Since K is assumed to act co-cubically, α is embedded in link(σ), except that its endpoints may have been identified, making it a loop. According to Lemma 4.12, α is a loop if and only if there is a CG(Y)-loop of the form p α .g that represents a conjugacy class in K. In particular, this condition only depends on the orbit α and not on α itself. We associate to α the element p = p α and the subgroup Q = Q t( α ) , as described in the preamble to this section.
Since X itself is a CAT(0) cube complex, the 1-cell α is not a loop. Applying Lemma 4.12 in case K = {1} we see that p ∈ Q. On the other hand, to say that p ∈ Q.K is the same as saying there is no CG(Y)-loop of the form p α .g which represents an element of K (since in such a CG(Y)-loop the element g must be in the local group G t( α ) ). This proves the result.
The next result is an application of Lemma 4.16 in case of edge-paths of length 2 consisting of 1-cells in different G-orbits (since then the K-non-backtracking condition is vacuous).
Theorem 5.7. Under Assumption 5.4 there exists a finite set F 2 ⊂ Q(G) 2 × G 2 so that for each (Q 1 , Q 2 , p 1 , p 2 ) ∈ F 2 we have and so that if (i) K G is co-cubical; and (ii) for each (Q 1 , Q 2 , p 1 , p 2 ) ∈ F 2 we have K ∩ p 1 Q 2 p 2 Q 2 = ∅ then every edge-loop of length 2 in a link in K X consists of 1-cells in the same G-orbit.
Proof. The proof is similar to the proof of Theorem 5.6 above. Lemma 4.16 implies that it is enough to verify that no link in a cube of K X contains a pair of 1-cells α and β in distinct G-orbits α , β so that there is a CG(Y)-loop p α · g 1 · p β · g 2 representing an element of K.
There are finitely many pairs of such orbits, and to each such pair we can associate the elements Since X is a CAT(0) cube complex, there are no non-backtracking edge-loops of length 2 in any links in X, so applying Lemma 4.16 with K = {1} we see that 1 ∈ p 1 Q 1 p 2 Q 2 . The result now follows from Lemma 4.16 with our choice of K.
For edge-paths of length 2 consisting of 1-cells in the same G-orbit, the condition is slightly more complicated, as K-backtracking edge-paths are possible.
Theorem 5.8. Under Assumption 5.4 there exists a finite set F 3 ⊂ Q(G) 2 × G 2 so that for each (Q 1 , Q 2 , p 1 , p 2 ) ∈ F 3 we have and so that if (i) K G is co-cubical; (ii) no link in K X contains an edge-loop of length 1; and (iii) for every (Q 1 , Q 2 , p 1 , p 2 ) ∈ F 3 we have )) = ∅ then no link in K X contains an immersed edge-loop of length 2 consisting of 1-cells in the same orbit.
Proof. Because of assumptions (i) and (ii) we only need to be concerned with the following situation: There is some cubeσ of X and some 1-cellα in its link so that the following hold.
(1) There is some g ∈ G so that g fixes t(α) but notα.
(2) There is some h ∈ G so that h(i(α)) = i(gα) but h −1 gα =α. There are finitely many orbits of pairs (σ,α) of this type. For each orbit we pick a representative, and describe an element of Q(G) 2 × G 2 as in the theorem. If Equation (6) is satisfied for this element, then K X will contain no immersed edge-loop of length 2 consisting of 1-cells in the orbit ofα.
We apply Lemma 4.16 to a path of length 2 of the form α.α where α is the image ofα in K X and α is the (oppositely oriented) image of a translate ofα by an element of the stabilizer ofσ. Any immersed loop of the type we are trying to rule out gives rise to a K-nonbacktracking CG(Y)-loop p α · g 1 · p α · g 2 representing a conjugacy class in K. We let p 1 = l p α , p 2 = l p α , Q 1 = Q t( α ) and Q 2 = Q i( α ) . Using Lemma 4.15, the loop p α · g 1 · p α · g 2 is K-nonbacktracking if and only if g 1 / ∈ E α K t( α ) and g 2 / ∈ E α K i( α ) . The subgroup of Q 1 corresponding to E α is equal to Q 1 ∩ Q p2 2 , and the subgroup of Q 2 corresponding to E α is Q 2 ∩ Q p1 1 . Thus an element p 1 q 1 p 2 q 2 of p 1 Q 1 p 2 Q 2 comes from a K-nonbacktracking CG(Y)loop if and only if q 1 / ∈ Q p2 2 (K ∩ Q 1 ) and q 2 / ∈ Q p1 1 (K ∩ Q 2 ). Applying Lemmas 4.15 and 4.16 in case K = {1} and K X = X is CAT(0), we see that our tuple satisfies Equation (5). For an arbitrary K we see that when Equation (6) is satisfied, there is no immersed edge-loop of length 2 in a link in K X consisting of images of translates ofα.
In order to apply Lemma 4.17, in each of the following two results we make the extra assumption that K is so that no link in K X contains an edge-loop of length 1 or 2. The following result is a translation of Lemma 4.18.
and so that if (i) K G is co-cubical; (ii) no link in K X contains an edge-loop of length 1 or 2; and (iii) for all (Q 1 , Q 2 , Q 3 , p 1 , p 2 , p 3 ) ∈ F 4 we have then every edge-loop of length 3 in a link of K X has image in Y which bounds a 2-cell.
Proof. Condition (ii) and Lemma 4.17 imply that it suffices to consider immersed loops of length 3 in links in K X . For each choice of triple of G-orbits α , β , γ of 1-cells in links in X whose image in Y forms a loop, but whose image does not bound a 2-cell in Y (in the sense described at the beginning of this subsection), we proceed as follows. We associate the elements p 1 = l p α , p 2 = l p β , p 3 = l p γ , Since X is a CAT(0) cube complex, we can apply Lemma 4.18 to see that Now let K G be co-cubical, and satisfy conditions (i)-(iii) from the statement. Condition (iii) implies that condition (2) from Lemma 4.18 does not hold, and by that lemma there is no immersed loop of length 3 in a link in K X whose image in Y is α , β , γ .
Since there are finitely many such triples α , β , γ , the theorem follows.
Finally, we deal with edge-loops of length 3 in links in K X whose image in Y does bound a 2-cell.
Using this terminology, we have the following translation of Proposition 4.26.
Theorem 5.11. Under Assumption 5.4 there exists a finite set F 5 ⊆ Q(G) 3 × G 6 so that for each A = (Q 1 , Q 2 , Q 3 , p 1 , p 2 , p 3 , h 1 , h 2 , h 3 ) we have and so that if (i) K G is co-cubical; (ii) no link in K X contains an edge-loop of length 1 or 2; and (iii) for all (Q 1 , Q 2 , Q 3 , p 1 , p 2 , p 3 , h 1 , h 2 , h 3 ) ∈ F 5 we have then no link in K X contains an edge-loop of length 3 which does not bound a 2-cell but whose image in Y bounds a 2-cell.
Proof. For each choice of triple of orbits α , β , γ whose image in Y bounds a 2-cell (in the sense described at the beginning of this subsection), we proceed as follows. Without loss of generality we choose representatives α, β, γ of these orbits so that there is a 2-cell τ with boundary α · β · γ. We associate the elements Once again, since X is a CAT(0) cube complex, we can apply Proposition 4.26 to see that When the conditions g 1 ∈ E αi g τi,αi E β K t( αi ) , etc. from the statement of Proposition 4.26 are translated into statements about the group G we get exactly g 1 ∈ B 1 (K ∩ Q 1 ), etc., which gives the statement in the conclusion of the result.
Since there are finitely many such triples α , β , γ , the theorem follows.

Dehn filling
In this section we prove some results about group-theoretic Dehn filling. Theorem 6.4 gives a 'weak separability' of certain multi-cosets, and generalizations of multi-cosets, and is used to find subgroups K which satisfy the conditions from Theorems 5.6-5.11. Theorem 6.4 may be of independent interest, and we expect it to have applications beyond the scope of this paper. The second main result of this section is Theorem 6.8, from which Theorem F from the introduction follows quickly by induction.
6.1. Dehn fillings. Let (G, P) be a group pair, and let N = {N P P | P ∈ P} be a choice of normal subgroups of the peripheral groups. The collection N determines a (Dehn) filling (G, P) of (G, P), where G = G/K for K the normal closure of N , and P equal to the collection of images of elements of P in G. The elements of N are called filling kernels. We sometimes write such a filling using the notation π : (G, P) → (G, P), omitting mention of the particular filling kernels.
If N P< P (i.e. N P is finite index in P ) for all P ∈ P, we say that the filling is peripherally finite. If H < G and for all g ∈ G, |H ∩P g | = ∞ implies N g P ⊆ H, then the filling is an H-filling. If H is a family of subgroups, the filling is an H-filling whenever it is an H-filling for every H ∈ H.
A property P holds for all sufficiently long fillings of (G, P) if there is a finite set S ⊆ P \ {1} so that P holds whenever ( N ) ∩ S = ∅. It is frequently useful to restrict attention to specific types of fillings (peripherally finite, H-fillings, etc.). If A is a property of fillings we say that P holds for all sufficiently long A-fillings if, for all sufficiently long fillings, either P holds or A does not hold.

6.2.
Relatively hyperbolic group pairs. We refer the reader to [16] for a background on relatively hyperbolic groups. In that paper, given a group pair (G, P) (consisting of finitely generated groups) a space called the cusped space is built, which is δ-hyperbolic (for some δ) if and only if (G, P) is relatively hyperbolic. See [16,Section 3] for the construction and basic geometry of the cusped space. The following result is essentially contained in [7,Theorem 7.11].
Theorem 6.1. Suppose that G is a hyperbolic group and that P is a finite collection of subgroups of G. Then (G, P) is relatively hyperbolic if and only if P is an almost malnormal family of quasi-convex subgroups.
Recall that P = {P 1 , . . . , P n } is almost malnormal if whenever P i ∩ P g j is infinite, we have i = j and g ∈ P i .
We can use the notion of height (see Definition 3.41) to measure how far away a family of subgroups is from being almost malnormal.
We now define the induced peripheral structure on G associated to a finite collection of quasi-convex subgroups of a hyperbolic group, in analogy with the construction from [2, Section 3.1]. We remark that the fact that there is a bound on the number k of g i H i as above follows from Proposition 3.42.
The following can be proved in the same way as [2, Proposition 3.12].
Lemma 6.3. Suppose that G is hyperbolic and H is a finite collection of quasiconvex subgroups of G.
(1) The induced peripheral structure P is a finite collection of groups. The pair (G, P) is relatively hyperbolic. The definition we use for relatively quasi-convex is that from [2]. In [22,Appendix A] it is proved that this is the same notion as the various notions defined in [20]. A subgroup H is full if whenever P is a parabolic subgroup so that H ∩ P is infinite we have H ∩ P< P .
6.3. The appropriate meta-condition. The goal of this subsection is to prove Theorem 6.4 below. The special case that n = 1 and S 1 = ∅ is [2,Proposition 4.5], which is about keeping elements out of full quasi-convex subgroups when performing long Dehn fillings. Here we generalize to multi-cosets of full quasiconvex subgroups, possibly with some elements deleted. Although the present result is more general, our proof is simpler, using the more appealing "Greendlinger Lemma"-type Theorem 6.6 below in place of the somewhat technical [2, Lemmas 4.1 and 4.2]. 4 Theorem 6.4. Let (G, P) be relatively hyperbolic, and let Q be a collection of full relatively quasi-convex subgroups. For 1 ≤ i ≤ n, let p i ∈ G, Q i ∈ Q and S i ⊆ Q i be chosen to satisfy: Then for sufficiently long Q-fillings G → G/K, the kernel K contains no element of the form The five conditions in the conclusions of Theorems 5.6 -5.11 each fall into the scheme of the conditions in Theorem 6.4. Therefore, we may apply Theorem 6.4 to obtain the following result. We remark that the following result is stated in the generality of relatively hyperbolic groups acting cocompactly on cube complexes with full relatively quasi-convex subgroups. This is greater generality than is strictly required for the proof of Theorem F. However, we believe that this extra generality will be of use in future work, and should be of independent interest. Corollary 6.5. Suppose that (G, P) is relatively hyperbolic and that G acts cocompactly on the CAT(0) cube complex X. Suppose every parabolic element of G fixes some point of X, and that cell stabilizers are full relatively quasi-convex. Let σ 1 , . . . , σ k be representatives of the G-orbits of cubes of X. For each i let Q i be the finite index subgroup of Stab(σ i ) consisting of elements which fix σ i pointwise. Let For sufficiently long Q-fillings of (G, P), with kernel K, the quotient K X is a CAT(0) cube complex.
Proof. The kernels of Dehn fillings are always generated by parabolic elements, and the parabolic elements act elliptically by assumption. Thus the kernel of any Dehn filling is generated by elliptic elements, so K X is simply-connected by Theorem 4.1. For sufficiently long Q-fillings the fact that G σi ∩ K ≤ Q i follows from [2,Proposition 4.4], so by Proposition 4.3 for such fillings K X is a cube complex. Therefore, we may assume that the subgroup K is co-cubical (in the sense of Terminology 5.5). It remains to show that for sufficiently long Q-fillings K X is non-positively curved. It follows from Theorems 5.6-5.8 and 6.4 that for sufficiently long Q-fillings each link of each cell in K X is simplicial. Thus it follows from Theorem 5.9, 5.11 and 6.4 that for sufficiently long Q-fillings, each link of each cell in K X is also flag, which means that K X is non-positively curved by Theorem 4.5.
To prove Theorem 6.4, we use the following "Greendlinger Lemma" (cf. [15,Lemma 2.41]): Theorem 6.6. Let C 1 , C 2 > 0. Suppose that (G, P) is relatively hyperbolic, with cusped space X. For all sufficiently long fillings G → G/K, and any geodesic γ in X joining 1 to g ∈ K \ {1}, there is a horoball A so that (1) γ penetrates A to depth at least C 1 , and (2) there is an element k of K stabilizing A, so that, for two points a, b in A and lying on γ at depth at least C 1 , d(a, kb) < d(a, b) − C 2 (in particular d(1, kg) < d(1, g) − C 2 ).
Proof. Let δ > 0 be such that X is δ-hyperbolic, and so are the cusped spaces for sufficiently long fillings (that there exists such a δ is [2, Proposition 2.3]). We only consider such fillings, without further mention of this assumption. Now choose L, so that every L-local (1, C 2 )-quasi-geodesic lies within anneighborhood of any geodesic with the same endpoints. (Such L, only depend on δ and C 2 . See [10,Ch. 3].) Now choose a filling long enough so that every (2L + C 1 + 2 )-ball centered on the Cayley graph embeds in the quotient cusped space. Let K be the kernel of the filling, and choose g ∈ K \ {1}. Let γ be a geodesic from 1 to g; let γ be the projection to the cusped space X/K for G/K. Within an (L + C 1 + 2 )neighborhood of the Cayley graph, γ is an L-local geodesic. But γ cannot be an L-local (1, C 2 )-quasi-geodesic everywhere, since it is a loop with diameter larger than .
In particular, there is a subsegment σ of γ of length l ≤ L so that the endpoints a and b of σ are less than l − C 2 apart. This subsegment σ must moreover lie in the image of a single horoball.
The corresponding points a and b on γ lie at depth at least C 1 in a horoball A of X. Since d(a, b) < l − C 2 , there is some element k ∈ K stabilizing A so that d(a, kb) < l − C 2 , as desired.
The following result follows immediately from [22, A.6].
Lemma 6.7. Suppose that (G, P) is relatively hyperbolic with cusped space X and that (H, D) ≤ (G, P) is a full relatively quasi-convex subgroup. There exists a constant κ satisfying the following: Suppose that g ∈ G and that x 1 , x 2 ∈ gH. Suppose that γ is a geodesic in X between x 1 and x 2 . Further, suppose that aP (for a ∈ G and P ∈ P) is a coset so that γ intersects the horoball corresponding to aP to depth at least κ. Then P is infinite and P a ∩ H g has finite-index in P a .
Proof of Theorem 6.4. Let X be the cusped space associated to (G, P) and suppose that X is δ-hyperbolic. Let C 2 be any positive number, and let C 1 = max{|p i |, κ}+ 2(n + 100)δ, where κ is the constant from Lemma 6.7 above. Suppose that K is the kernel of a filling which is long enough to satisfy the conclusion of Theorem 6.6 with these constants.
In order to obtain a contradiction, suppose that there is an element g ∈ K which is of the form g = p 1 t 1 · · · p n t n , where t i ∈ Q i \ ((K ∩ Q i )S i ), and suppose that g is chosen so that d X (1, g) is minimal amongst all such choices.
Since for each i we have Q i \ ((K ∩ Q i )S i ) ⊆ Q i \ S i , the assumption of the theorem implies that g = 1. We can represent the equation g = p 1 t 1 · · · t n p n by a geodesic (2n + 1)-gon in X, joining the appropriate elements of the Cayley graph in turn by X-geodesics. Let γ be the geodesic for g, ρ i the geodesic for p i and τ i the geodesic for t i .
Since g ∈ K \ {1}, by Theorem 6.6 there exist a horoball A in X, an element k ∈ K stabilizing A, and points a, b on γ at depth at least C 1 so that k stabilizes A and d(a, kb) < d(a, b) − C 2 . In particular, we have d(x, kgx) < d(x, gx) − C 2 . The geodesic (2n + 1)-gon is (2n − 1)δ-thin, so b lies within distance (2n − 1)δ of some side other than γ. The paths ρ i do not go deeply enough into any horoballs to be this close to b, so b lies within (2n − 1)δ of some point b on some τ i . By the choice of C 1 , b lies at depth at least κ in A.
Write A = aP for some P ∈ P. Note that τ i is a geodesic between two points in the coset p 1 t 1 · · · p i Q i . By Lemma 6.7, P a ∩ Q p1t1···pi i has finite-index in P a . Since the filling is a Q-filling, we have that k ∈ Q p1t1···pi i . Let k = k (p1t1···pi) −1 , and let t i = k t i . Then k ∈ K ∩ Q i . Note that kg = p 1 t 1 · · · p i (k t i )p i+1 · · · p n t n . Since t i ∈ (K ∩ Q i )S i , we have that t i ∈ (K ∩ Q i )S i . Therefore, the element kg is another element of the required form, contradicting the choice of g as the shortest such. This completes the proof of Theorem 6.4.
For the remaining properties, we show that they hold for sufficiently long peripherally finite Q-fillings of (G, P). Therefore, to ensure that all of the properties hold, it suffices to take a sufficiently long Q-filling with each N i<Ṗi .
Property (3) holds automatically for any Q-filling, since K P is generated by conjugates of elements in Q, and each such conjugate lies in a cell stabilizer.
We now explain how to ensure each of the remaining properties in turn for sufficiently long Q-fillings.
For property (4), suppose that F i {1} is a set of coset representatives for Q i in Stab(σ i ). To ensure that (4) holds, it suffices to keep (the image of) each element of F i out of the image of Stab(σ i ) in G. This is true for sufficiently long Q-fillings by [1,Theorem A.43.4], because Q i has finite index in Stab(σ i ).
Property (5) holds for sufficiently long peripherally finite Q-fillings of (G, P) by an entirely analogous argument to that of [1,Theorem A.47].
The group G as above acts isometrically on X = K P X with quotient naturally isomorphic (as a topological space, but not as a complex of groups) to G X . Therefore, if the action of G on X is not proper, we can apply Theorem 6.8 to this action, to obtain a further quotient. By induction on height, we obtain the following result from the introduction.
Theorem F. Suppose that the hyperbolic group G acts cocompactly on a CAT(0) cube complex X and that cell stabilizers are virtually special and quasi-convex. There exists a quotient G = G/K so that (1) The quotient K X is a CAT(0) cube complex; (2) The group G is hyperbolic; and (3) The action of G on K X is proper (and cocompact).

Appendix A. A quasi-convexity criterion
In this appendix, we give a criterion (Theorem A.3) for a possibly infinite union of quasi-convex sets in a hyperbolic space to be quasi-convex. This criterion is used in the forward direction of Theorem A: quasi-convex cell stabilizers imply quasi-convex hyperplane stabilizers. This criterion may be of independent interest.
Since any subset is a union of points, clearly some assumptions are needed. We begin with a basic lemma about finite unions of quasi-convex subsets.
Proof. Consider a pair of points x ∈ P r , y ∈ P s . Without loss of generality, assume that r < s (the case r = s being straightforward). Now choose a sequence of points p i ∈ P i ∩ P i+1 for r ≤ i < s, let σ be a geodesic between x and y and let u be a point on σ. Our task is to bound the distance from u to P .
Consider the geodesic polygon with one side the geodesic σ = [x, y] and the other sides the geodesics forming γ. Let r 0 = r+s triangle σ, [x, p r0 ], [p r0 , y]. By δ-hyperbolicity, u lies within δ of one of [x, p r0 ] and [p r0 , y]. Suppose it is [x, p r0 ] (the other case being entirely similar), and suppose that u 1 ∈ [x, p r0 ] is within δ of u. Now let r 1 = r+r0 2 and consider the geodesic triangle [x, p r1 ], [p r1 , p r0 ], [p r0 , x]. By δ-hyperbolicity, u 1 is within δ of one of [x, p r1 ], [p r1 , p r0 ], so there is u 2 on one of these sides within δ of u 1 and within 2δ of u.
We proceed in this manner, in each case making the interval of indices half as long. After t steps of this argument we find a point u t which is within within tδ of u.
After at most d = log 2 (k) + 1 steps, we have a geodesic triangle where two sides are [p l , p l+1 ], [p l+1 , p l+2 ] (or maybe one endpoint x or y), and we have u d within dδ of u, but also within of P . This proves the lemma.
The following straightforward instance of "linear-beats-log" is tailored for use in the proof of Theorem A.3.
Lemma A.2. Fix δ, > 0, and let g(x) = δ (log 2 (x + 1) + 1) + . For any m > 0 and c ≥ 0 there exists a natural number R m, ,δ so that for all R 0 > R m, ,δ , we have The next result states that under appropriate hypotheses, the union of an arbitrary number of quasi-convex subsets is itself quasi-convex, with constant not depending on the number of such subsets.
Theorem A.3. Suppose that Υ is a δ-hyperbolic space and that m, > 0 and c ≥ 0 are real numbers. There exists a constant so that for any (finite or countably infinite) collection of subsets {X i } Λ i=1 of Υ for which (1) Each X i is -quasi-convex; (2) For each i we have X i ∩ X i+1 = ∅; and (3) For any i, j, if x ∈ X i and y ∈ X j we have d(x, y) ≥ m (|i − j| − c), the set X = ∪X i is -quasi-convex.
On the other hand, suppose that Λ > 100R and fix u, v ∈ X. Let j, k be so that u ∈ X j , v ∈ X k and without loss of generality suppose that j ≤ k. It suffices to show that any geodesic [u, v] stays uniformly close to X j ∪· · ·∪X k . If |k −j| ≤ 100R then this follows from Lemma A.1, so suppose that |k−j| > 100R. Let Y = X j ∪· · ·∪X k .
Our strategy is to build a path between u and v which is (i) uniformly quasigeodesic; and (ii) stays uniformly close to Y . The theorem then follows by quasigeodesic stability. Choose a sequence of indices t 0 = j, t 1 , . . . , t s−1 , t s = k so that for each 0 ≤ r ≤ s − 2 we have t r+1 − t r = 100R, and t s − t s−1 ∈ Z ∩ [100R, . . . , 200R] . Moreover, for each 0 ≤ r ≤ s choose some u r ∈ X tr . We require u 0 = u and u s = v.
Since we assume R ≥ 1 we know that K > δ.
Since we know that for each r ∈ {0, . . . , s − 1} we have t r+1 − t r ≤ 200R we know that the set is K-quasi-convex, by Lemma A.1. In particular the geodesic γ r lies in a Kneighborhood of Y r .
For each r ∈ {0, . . . , s − 1} and each x ∈ γ r , let π r (x) denote the set of closest points on Y r to x. Furthermore, let I r (x) be the set of indices l so that π r (x) ∩ X l = ∅.
Claim A.3.1. For any v ∈ {t r , . . . , t r+1 } there exists x v ∈ γ r so that Proof of Claim A.3.1. For any y ∈ π r (x) we have d(x, y) ≤ K. Now, if x and x are adjacent vertices and y ∈ π r (x) with y ∈ X k and z ∈ π r (x ) with z ∈ X l then m(|k − l| − c) ≤ d(y, z) ≤ d(y, x) + d(x, x ) + d(x , z) ≤ 2K + 1, so |k − l| ≤ 2K+1 m + c. The claim now follows immediately from the fact that t r ∈ I r (u r ) and t r+1 ∈ I r (u r+1 ), letting x and x run over adjacent pairs of vertices in γ r . This finishes the proof of Claim A.3.1.
Suppose 0 ≤ r ≤ s − 1. Using Claim A.3.1, we can choose a point x r ∈ γ r and a point y r ∈ π r (x r ) so that y r ∈ X kr and ( †) k r − t r + t r+1 2 ≤ 2K + 1 2m + c.
Now, for each r ∈ {1, . . . , s − 1}, let σ r be a geodesic between y r−1 and y r . Further, let σ 0 be a geodesic from u to y 0 and let σ s be a geodesic from y s−1 to v (note that there is no point y s ). See Figure 2. We bound the Gromov product between σ t and σ t+1 for each t. (There is no reason to expect such a bound on the Gromov product between γ r and γ r+1 .) Though we have no control on the lengths of the segments σ 0 and σ s , the lengths of the other segments can be bounded below: Claim A.3.2. Suppose 0 < r < s. The length of σ r is at least 200K.
Proof. By the choice of the index k r in Equation ( †) we have (the equality follows from the choice of t r ). Below, we apply Lemma A.2 with R 0 = 200R, noting that K = g(200R), where g is the function from that lemma. We have The second inequality above follows from the fact that y i ∈ X ki so such points are at least distance m(k r − k r−1 − c) apart. The final inequality follows from the promised use of Lemma A.2. This completes the proof of Claim A.3.2.
Proof. We first handle the case that 0 < r < s − 1.
For i ∈ {r, r + 1}, the path σ i is one side of a pentagon. The other sides are (A) two sides of length at most K at either end of σ i , and (B) two 'halves' of adjacent geodesics: the second 'half' of γ i−1 and the first 'half' of γ i , joined at u i . See Figure  3.  By Claim A.3.2 the geodesics σ r and σ r+1 have length at least 200K. Let z be the point on σ r at distance exactly 8K from y r .
Since geodesic pentagons are 3δ-slim, we know that z must be distance at most 3δ from some point on one of the other four sides. However, it cannot be within distance 3δ of the geodesic between x r and y r since that geodesic has length at most K. Similarly, since |σ r | ≥ 200K, z cannot be within 3δ of the geodesic between x r−1 and y r−1 . We claim that z also cannot be within 3δ of the part of γ r−1 contained in the pentagon.
Indeed, suppose w ∈ γ r−1 , and choose i w ∈ I r−1 (w) ⊂ [t r−1 , t r ]. There is a point w of π r−1 (w) in X iw ; thus d(w, w ) ≤ K. The point x r is likewise within K of some X kr where k r satisfies the inequality ( †). This implies that and so using Lemma A.2 again. But this contradicts d(x r , w) ≤ d(x r , z) + d(z, w) ≤ 9K + 3δ ≤ 12K. We have shown that there is some point w on γ r between u r and x r within 3δ of z. Note that d(x r , w) ≥ d(y r , z) − K − 3δ ≥ 4K, since K ≥ δ. Now consider the pentagon formed with σ r+1 on one side, and the point z on σ r+1 which is distance exactly 8K from y r . An entirely analogous argument to the above shows that there is some w between x r and u r+1 on γ r so that d(z , w ) ≤ 3δ, and d(x r , w ) ≥ 4K. Since γ r is geodesic, we have d(w, w ) = d(w, x r ) + d(x r , w) ≥ 8K.
The cases r = 0 and r = s − 1 are symmetric, so it suffices to handle the case r = 0. See Figure 4. We are trying to show that (u, y 1 ) y0 ≤ 8K, so we may suppose without loss of generality that d(y 0 , u) > 8K. Thus there is a point z on σ 0 at distance exactly 8K from y 0 . Since d(x 0 , y 0 ) ≤ K, this point is within δ of a point w on γ 0 between u and x 0 .
For the point z on σ 1 at distance 8K from y 0 , we argue as before. We are again able to deduce that d(z, z ) > δ, and so (u, y 1 ) y0 ≤ 8K.
Thus, we have a collection of arcs σ i which form a broken geodesic between u and v with segments of length at least 200K (except possibly the first and last) and all Gromov product at most 8K at the corners. Thus the union of the σ i forms a global quasi-geodesic with uniformly bounded parameters. However, each σ i lies within a (3δ + K)-neighborhood of the union of the γ i , which in turn lie in a K-neighborhood of the union of the X i . As explained above, this suffices to prove that the union of the X i is -quasi-convex with the constant depending on the quantities δ, m, and , but not on the number of the X i , as required. This completes the proof of Theorem A.3.