Non-quasiconvex subgroups of hyperbolic groups via Stallings-like techniques
aa r X i v : . [ m a t h . G R ] F e b NON-QUASICONVEX SUBGROUPS OF HYPERBOLIC GROUPSVIA STALLINGS-LIKE TECHNIQUES
PALLAVI DANI AND IVAN LEVCOVITZ
Abstract.
We provide a new method of constructing non-quasiconvex sub-groups of hyperbolic groups by utilizing techniques inspired by Stallings’ fold-ings. The hyperbolic groups constructed are right-angled Coxeter groups(RACGs for short) and can be chosen to be 2-dimensional. More specifically,given a non-quasiconvex subgroup of a, possibly non-hyperbolic, RACG, ourconstruction gives a corresponding non-quasiconvex subgroup of a hyperbolicRACG.We use this construction to find non-quasiconvex subgroups of hyperbolicRACGs that are generated by words of length 2. Additionally, we explicitlyconstruct non-quasiconvex, finitely generated, free subgroups of hyperbolic,one-ended RACGs such that the intersection of such a subgroup with anyfinite-index subgroup of the RACG is not normal in the finite-index subgroup.We give examples of non-locally quasiconvex hyperbolic groups that are funda-mental groups of square complexes of nonpositive sectional curvature. Finally,we apply our method to explicitly construct hyperbolic RACGs containingfinitely generated subgroups which are not finitely presentable as well as hy-perbolic RACGs which have subgroups that are not quasiconvex and containclosed surface subgroups. Introduction
Given a group G with generating set T , a subgroup H of G is quasiconvex (withrespect to T ) if some finite neighborhood of H in the Cayley graph C of ( G, T )contains every geodesic connecting a pair of vertices in H ⊂ C . If G is hyperbolic,then whether or not H is quasiconvex does not depend on the finite generating setchosen. In this article, we provide a simple construction of finitely generated non-quasiconvex subgroups of hyperbolic groups using techniques inspired by Stallings’foldings. The hyperbolic groups we construct are additionally right-angled Coxetergroups and can be chosen to be 2-dimensional.Given a simplicial graph Γ with vertex set V and edge set E , the corresponding right-angled Coxeter group (RACG for short) W Γ is the group with presentation: h s ∈ V | s = 1 for all s ∈ V, st = ts for all ( s, t ) ∈ E i We refer to the generators in this presentation as the standard generating set for W Γ , and quasiconvexity of subgroups of W Γ will always be assumed to be withrespect to this generating set. We say that W Γ is 2 -dimensional if Γ does notcontain a 3-cycle (correspondingly, its associated Davis complex is a CAT(0) cubecomplex that is at most 2-dimensional).Let W Γ be a RACG and H < W Γ a subgroup generated by a finite set of words T in W Γ . Let R be the “rose graph” associated to T , i.e., a bouquet of | T | circlesin which each circle is subdivided and labeled by a word in T . In [DL19], we define a completion Ω of H as the direct limit of a sequence of fold, cube attachmentand cube identification operations which are performed on R . We show in thatarticle that many properties of H < W Γ are reflected by those of Ω. Notably, H isquasiconvex in W Γ if and only if Ω is finite.Given any simplicial graph Γ, we define in this article the notion of Γ-partitegraphs. These graphs have a large-scale structure which reflects the graphicalstructure of Γ. We show that given any graph Γ, there are infinitely many Γ-partitegraphs ∆ so that W ∆ is hyperbolic and 2-dimensional.The importance of Γ-partite graphs in our setting is explained by the followingconstruction. Given a Γ-partite graph ∆, and a subgroup H < W Γ generated by afinite set of words T in W Γ , we define a corresponding finite set of words T in W ∆ which generates a subgroup H < W ∆ . Moreover, fold and cube attachment opera-tions that are performed on the rose graph associated to T have direct analoguesto operations performed on the rose graph associated to T . In fact, we show thereis a finite-to-one map from the 1-skeleton of a completion of H to the 1-skeleton ofa completion of H that is defined by simply collapsing certain bigons to edges. Inparticular, we deduce that H is quasiconvex in W Γ if and only if H is quasiconvexin W ∆ . Using this construction, we show: Theorem A.
Given any finitely generated non-quasiconvex subgroup H of a (possi-bly non-hyperbolic) RACG W Γ , there exist infinitely many Γ -partite graphs ∆ suchthat W ∆ is hyperbolic, -dimensional and contains the non-quasiconvex subgroup H . Moreover, if H is torsion-free (resp. not normal in W Γ ) then H is torsion-free(resp. not normal in W ∆ ). Non-quasiconvex subgroups of non-hyperbolic
RACGs are plentiful. They can,for instance, be found in RACGs that split as a product of infinite groups. Conse-quently, our construction allows for many easy, explicit examples of non-quasiconvexsubgroups of hyperbolic RACGs.As a simple application, we construct explicit non-quasiconvex subgroups ofhyperbolic RACGs given by short generators:
Theorem B.
There are hyperbolic, -dimensional RACGs which contain a finitelygenerated, non-quasiconvex subgroup whose generators are all length words in thestandard generating set of the RACG. As any subgroup of a RACG that is generated by length one words must beconvex (and therefore quasiconvex), the above theorem is in some sense optimal.We also give many explicit examples of finitely generated, free, non-quasiconvexsubgroups of RACGs. Moreover, these subgroups are not the kernel of any homo-morphism with domain a finite index subgroup of the RACG:
Theorem C.
There are hyperbolic, -dimensional, one-ended RACGs which con-tain a finitely generated, non-quasiconvex, free subgroup F such that, for any finite-index subgroup K of the RACG, F ∩ K is not normal in K . An advantage of our construction of non-quasiconvex subgroups is that, if theyare torsion-free, then the fundamental group of their completion is isomorphic tothe subgroup. Moreover, in practice, these completions often have an explicit de-scription, being analogues of well-understood, simple completions of subgroups ofnon-hyperbolic groups. We utilize this to prove that the subgroups in the previoustheorem are free, by applying Van-Kampen’s theorem to their completions.
ON-QUASICONVEX SUBGROUPS 3
Wise defines a notion of sectional curvature for 2-complexes [Wis04] and showsthat the fundamental group of a compact Euclidean 2-complex with negative sec-tional curvature is always locally quasiconvex (i.e., all finitely generated subgroupsare quasiconvex) and is consequently coherent (i.e., all finitely generated subgroupsare finitely presented). On the other hand, fundamental groups of 2-complexeswith nonpositive sectional curvature provide a fringe case where coherence is con-jectured [Wis19, Conjecture 12.11] and yet local quasiconvexity can fail. We giveexplicit examples (see Theorem 5.3) of this phenomenon by constructing hyperbolicgroups that are fundamental groups of right-angled square complexes of nonpos-itive sectional curvature and which contain a non-quasiconvex, finitely generatedfree subgroup.In Theorem 6.3, we also demonstrate how our method can be used to explicitlyconstruct finitely generated, not finitely presentable subgroups of hyperbolic groupsand non-quasiconvex subgroups containing closed surface subgroups.There are other, but not many, known methods of constructing non-quasiconvexsubgroups of hyperbolic groups, and we give a non-extensive summary of such con-structions. Many of the known examples come from infinite-index, infinite, normalsubgroups of hyperbolic groups which are known to always be non-quasiconvex[ABC + Z . Thismethod is utilized, for instance, in [Bra99] to give examples of non-hyperbolic,finitely presented subgroups of hyperbolic groups. Bestvina-Brady Morse theoryis also applied to RACGs in [JNW19] giving another source of examples in thesegroups. In [Kap99] it is also shown that any non-elementary, torsion-free, hyperbolicgroup appears as a non-quasiconvex subgroup of a different hyperbolic group.This article is organized as follows. With our setting in mind, in Section 2 wegive a brief background on completions and nonpositive sectional curvature. Wedefine Γ-partite graphs in Section 3, and we show that such graphs can always beconstructed to have some additional desirable properties. Our main constructionand results (including Theorem A) are given in Section 4. Moreover, Theorem B isgiven as an example in this section. We prove Theorems C and give nonpositive sec-tional curvature examples in Section 5. Finally, in Section 6 we show how to applyour method to construct hyperbolic RACGs which contain explicit finitely gener-ated, not finitely presentable subgroups and non-quasiconvex subgroups containingclosed surface subgroups. Acknowledgements.
IL would like to thank Michah Sageev for helpful conversa-tions. The authors thank Jason Behrstock and Mahan Mj for comments on a draftof this paper. PD was supported by NSF Grant No. DMS-1812061, and IL wassupported in part by a Technion fellowship.
PALLAVI DANI AND IVAN LEVCOVITZ Background
Completions.
We refer the reader to [DL19] for details regarding the con-structions and results in this section.A cube complex is a cell complex whose cells are Euclidean unit cubes (of anydimension). Given a graph Γ, a cube complex is Γ -labeled if its edges are labeledby vertices of Γ. All Γ-labeled cube complexes considered in this article have theproperty that edges on opposite sides of squares have the same label. In particular,given the labels of d edges which are all incident to a common vertex and containedin a common d -cube, the labels of the remaining edges of this d -cube are completelydetermined. This will often be used without mention.By a path in a cube complex Σ, we shall mean a simplicial path in the 1-skeletonof Σ. The label of such a path is a word s . . . s m in V (Γ) such that s i is the labelof the i th edge traversed by Σ.Given a Γ-labeled cube complex Σ, we define three operations, each of whichproduces a new Γ-labeled cube complex: Fold operation:
Let e and f be distinct edges of Σ which are incident to acommon vertex and have the same label. A fold operation produces a new complexin which e and f are identified to a single edge with the same label as e and f . Cube identification operation:
Given two or more distinct d -cubes in Σ withcommon boundary, a cube identification operation identifies these cubes. Cube attachment operation:
Let e , . . . , e d , with d ≥
2, be edges of Σ allincident to a common vertex, with distinct labels corresponding to a d -clique of Γ.A cube attachment operation attaches a labeled d -cube c to Σ by identifying a setof d edges of c , all incident to a common vertex, to the edges e , . . . , e d . Note thatthe labels of edges of c are completely determined by the labels of e , . . . , e d .A Γ-labeled cube complex is folded if no fold or cube identification operationscan be performed on it. Such a complex is cube-full if given any set of d ≥ d -clique of Γ, itfollows that these edges are contained in a common d -cube.Let Σ be a Γ-labeled cube complex. Let Ω , Ω , . . . be a possibly infinite sequenceof cube complexes such that Ω = Σ and, for i >
0, the complex Ω i is obtainedfrom Ω i − by performing one of the three above operations. There is a natural map f i : Ω i → Ω i +1 . Let Ω be the direct limit of this sequence of complexes. We saythat Ω is a completion of Σ if Ω is folded and cube-full. If Σ is has finitely manycubes, one can always construct a completion of Σ [DL19, Proposition 3.3].Let H be a finitely generated subgroup of a RACG W Γ , generated by words T = { w , . . . , w m } in V (Γ). The rose graph associated to T is the Γ-labeled graphwith a base vertex B and a | w i | -cycle attached to B with label w i for each 1 ≤ i ≤ m . For this article, it suffices to define a completion of H with respect to T as acompletion Ω of the rose graph associated to T . The image of B in Ω is defined tobe the base vertex of Ω.We say that a word w in V (Γ) is reduced if it has minimal length out of allpossible words in V (Γ) representing the same element of W Γ as w . We summarizethe key properties of completions that we will need: ON-QUASICONVEX SUBGROUPS 5
Theorem 2.1. [DL19]
Let H be a subgroup of a RACG W Γ generated by a finiteset T of words in W Γ . Let Ω be a completion of H with respect to T . Then: (1) The completion Ω is finite if and only if H is quasiconvex in W Γ with respectto the standard generating set of W Γ . (2) The group H has torsion if and only if there is some loop in the -skeletonof Ω with label a reduced word whose letters are contained in a clique of Γ . (3) If H is torsion-free, then Ω is non-positively curved and has fundamentalgroup isomorphic to H . (4) Any loop in Ω based at B has label an element of H , and any reduced wordwhich is an element of H appears as the label of some loop based at B . (5) Suppose additionally that W Γ does not split as a product with a finite factor(equivalently, there does not exist a vertex of V (Γ) which is adjacent toevery other vertex of V (Γ) ). Then H is of finite index in G if and only ifa) Ω is finite, and b) for every vertex v of Ω and for every s ∈ V (Γ) , thereis an edge incident to v in Ω , which is labeled by s . Nonpositive sectional curvature.
We review the notion of nonpositive sec-tional curvature simplified according to our setting. For a more extensive back-ground we refer the reader to [Wis04] and [Wis19].Given a graph Θ with vertex set V and edge set E , we define κ (Θ) = 2 − | V | + | E | spur of Θ is an edge containing a vertex of valence one, and Θ is spurless if itdoes not contain any spurs.Let Σ be a right-angled square complex. We say that Σ has nonpositive sectionalcurvature if given any vertex v of Σ and any connected, spurless, non-discrete (i.e.,containing at least one edge) subgraph Θ of the link of v , it follows that κ (Θ) ≤ nonpositive sectional curvature , ifall connected, spurless, non-discrete subgraphs Θ of Γ satisfy κ (Θ) ≤ W Γ is the fundamental group of a right-angledsquare complex of nonpositive sectional curvature.3. Γ -partite graphs Let Γ be a graph with vertex set { s , . . . , s n } . We say that a simplicial graph ∆is Γ -partite if its vertex set can be partitioned into n non-empty, disjoint sets A , . . . , A n such that the following two conditions hold:(1) No two vertices of A i are adjacent.(2) For all 1 ≤ i < j ≤ n , if s i is adjacent to s j in Γ, then the subgraph of ∆induced by A i ∪ A j is connected. Otherwise, if s i is not adjacent to s j in Γthen there are no edges in ∆ between A i and A j .We call A , . . . , A n the decomposition of ∆. We say that ∆ has cycle connectors if,for every 1 ≤ i < j ≤ n , the subgraph of ∆ induced by A i ∪ A j is either empty ora cycle. Similarly, we say that ∆ has path connectors if, for every 1 ≤ i < j ≤ n ,the subgraph of ∆ induced by A i ∪ A j is either empty or a simple path.The main result of this section is that given any simplicial graph Γ, there is aΓ-partite graph ∆ that can be chosen so that W ∆ is hyperbolic and 2-dimensional. PALLAVI DANI AND IVAN LEVCOVITZ
Figure 3.1.
A graph Γ on the left and, on the right, a Γ-partitegraph with cycle connectors, no 3-cycles and no 4-cycles.
Theorem 3.1.
Given a finite simplicial graph Γ , there exist infinitely many Γ -partite graphs which do not contain any simple -cycles or -cycles. Additionally,these graphs can be constructed to have either cycle connectors or path connectors.Proof. Let Γ have E edges and let V (Γ) = { s , . . . , s n } . Let k be a number which isnot divisible by 3 and which is larger than 8(3 E ). We construct a Γ-partite graph ∆with decomposition A , . . . , A n such that | A i | = k for each 1 ≤ i ≤ n . We showthat the constructed graph has no 3-cycles or 4-cycles. We first construct ∆ to havecycle connectors.The vertex set of ∆ is A ∪ · · · ∪ A n . For each 1 ≤ i ≤ n , we fix a labeling a i , . . . , a ik − for the vertices of A i . To define the edge set of ∆, we arbitrarily labelthe edges of Γ as e , . . . , e E , and arbitrarily orient each such edge. Fix such anedge e p , and suppose e p is incident to the vertices s x and s y in Γ and is orientedfrom s x to s y . We describe the edges of ∆ between A x and A y by describing whichvertices of A y are adjacent to a given vertex of A x . Each vertex s xl ∈ A x is adjacentto exactly two vertices of A y , namely a yl +3 p and a yl +2(3 p ) (where the arithmetic ismodulo k ). This choice of edges for each edge of Γ defines ∆.The following properties now follow:(1) As k is not divisible by 3, it follows that 3 p and k are relatively prime.Consequently, the edges between A x and A y form a simple 2 k -cycle thatincludes every vertex of A x ∪ A y .(2) A vertex a yl ′ ∈ A y is adjacent to two vertices of A x : a xl ′ − p and a xl ′ − p ) .(3) If a xi and a xj are distinct vertices of A x which share a neighbor in A y , then | i − j | ≡ p (mod k ). Similarly, if a yi ′ and a yj ′ are distinct vertices of A y which share a neighbor in A x , then | i ′ − j ′ | ≡ p (mod k ).It readily follows that ∆ is Γ-partite (by construction) and that ∆ has cycleconnectors (by (1) above).We now check that there are no simple 4-cycles in ∆. First note that, for each1 ≤ i ≤ n , no simple 4-cycle has all its vertices in A i , as there are no edges in A i . Moreover, no 4-cycle has all its vertices in A i ∪ A j for some 1 ≤ i < j ≤ n asthe subgraph induced by A i ∪ A j either forms a simple 2 k -cycle with k > c with vertices in A i ∪ A i ∪ A i for some { i , i , i } ⊂ { , . . . , n } . Without loss of generality, we may assume that there areedges e p , between s i and s i , and e q , between s i and s i , in Γ. Moreover, we may ON-QUASICONVEX SUBGROUPS 7 assume that c contains two vertices a i u and a i v in A i . As c is a 4-cycle, a i u and a i v have a common neighbor in A i and a common neighbor in A i . However, by(3) above, we have that | u − v | ≡ p (mod k ) and | u − v | ≡ q (mod k ). This is acontradiction as p = q and k > | p − q | . Thus, such a cycle c cannot exist.The only possibility now is that there is some 4-cycle c with vertices in A i ∪ A i ∪ A i ∪ A i for some { i , i , i , i } ⊂ { , . . . , n } , such that for each 0 ≤ j ≤ c contains exactly one vertex in A i j and there is an edge e p j in Γ between s i j and s i j +1 (subscripts taken modulo 4).To see that such a cycle c cannot exist, we consider an arbitrary length 4 path γ in ∆ with startpoint and endpoint in A i , and exactly one vertex in each of A i , A i and A i . Let a i l ∈ A i be the startpoint of γ . It follows from the definition of edgesof ∆ and from (2) above that the endpoint a i l ′ of γ satisfies: l ′ ≡ l + X m =0 ǫ m p m (mod k )where ǫ m ∈ {− , − , , } for each 0 ≤ m ≤ P m =0 ǫ m p m = 0. Suppose otherwise, for a contradiction. Let C (resp. C ′ ) be the sum of the positive (resp. absolute values of the negative) termsof P m =0 ǫ m p m . It follows that C = C ′ . However, as the numbers p , p , p and p are all distinct and positive, this implies that we have two ways to represent thenumber C in base 3. This contradicts the basis representation theorem. We havethus shown that P m =0 ǫ m p m = 0.As | P m =0 ǫ m p m | < k by our choice of k and as P m =0 ǫ m p m = 0, it followsthat l ′ l (mod k ). Thus, a i l and a i l ′ are distinct vertices. We have then shownthat there cannot be any 4-cycle c as described above.A similar proof shows that ∆ does not contain 3-cycles. This completes theconstruction of the claimed Γ-partite graph ∆ with cycle connectors and no 3-cycles or 4-cycles. To produce a Γ-partite graph ∆ ′ with path connectors and no3-cycles or 4-cycles, we can, for each edge ( s i , s j ) of Γ, remove an arbitrary edgefrom the subgraph of ∆ induced by A i ∪ A j . (cid:3) Remark 3.2.
The proof of the above theorem gives an explicit, easily imple-mentable construction for ∆, not just an existential one.Recall that a RACG is not one-ended if and only if its defining graph is eithera clique, is disconnected, or is separated by a clique (see [Dav15]). From this andthe structure of Γ-partite graphs, we immediately conclude:
Lemma 3.3.
Let Γ be a finite simplicial graph, and let ∆ be a Γ -partite graph. If W Γ is one-ended, then so is W ∆ . Main Construction
We give our main construction and result (Theorem 4.5) in this section. Through-out this section, we fix a simplicial graph Γ with vertex set { s , . . . , s n } and aΓ-partite graph ∆ with decomposition A , . . . , A n .Let Π be a ∆-labeled cube complex. Given 1 ≤ i ≤ n and vertices u and v of Π,a generalized edge between u and v with label A i is a collection of | A i | edges between u and v whose labels are in bijection with the vertices of A i . Note that u and v could be the same vertex. PALLAVI DANI AND IVAN LEVCOVITZ
Given a Γ-labeled cube complex Σ, we construct an associated ∆-labeled cubecomplex Σ. We call Σ the generalization of Σ with respect to ∆, and we willalways use the bar notation to denote the generalization of a cube complex. Whenthere is no confusion regarding the Γ-partite graph ∆, we simply say that Σ is thegeneralization of Σ.The vertex set of Σ is in bijective correspondence with that of Σ. The edges ofΣ are defined as follows: for each edge e of Σ there is a generalized edge with label A i between the corresponding vertices of Σ, where s i is the label of e . All edges ofΣ are defined this way.Note that the 1-skeleton of Σ is isomorphic to that of Σ, after we appropriatelycollapse generalized edges with label A i to single edges labeled by s i , for each1 ≤ i ≤ n . We call this map Σ → Σ the collapsing map.A generalized d -cube in Σ is a set of 2 d generalized edges whose image under thecollapsing map is the 1-skeleton of a d -cube of Σ.Let c be a d -cube in Σ, with d ≥
2, whose edges are labeled by distinct vertices s i , . . . , s i d of Γ. Then, by construction, there is a generalized d -cube c in Σ withgeneralized edges labeled by A i , . . . , A i d . For each ( a , . . . , a d ) ∈ A i × · · · × A i d such that a , . . . , a d span a d -clique in ∆, the complex Σ contains a single d -cubewhose 1-skeleton is the corresponding subset of 2 d edges of c . Note that even ifsome other d -cube in Σ has the same boundary as c , there is only one d -cube inΣ for each such tuple. All d -cubes of Σ, with d ≥
2, are defined this way. Thiscompletes the construction of Σ. By construction, no cube identification operationscan be performed to Σ.The next lemma is the key observation that allows us to prove our main theorem.
Lemma 4.1.
Let Σ be a Γ -labeled cube complex, and let Σ be the ∆ -labeled cubecomplex which is its generalization. Let Ω be a completion of Σ . Then there is acompletion of Σ that is isomorphic to the generalization Ω of Ω .Proof. We first establish three claims, the first of which is obvious by construction.
Claim 4.2.
If a cube identification operation can be performed on Σ to obtain anew complex Θ, then Σ is isomorphic to Θ.
Claim 4.3.
If a fold operation can be performed on Σ to obtain a new complex Θ,then a sequence of fold and cube identification operations can be performed to Σto obtain Θ.Suppose that there are edges e and f in Σ, each incident to a common vertex v and each labeled by the same s i ∈ V (Γ). Then there are generalized edges e and f in Σ, each incident to a common vertex and each labeled by A i . By performing | A i | folds, we identify e and f into a single generalized edge labeled by A i . Next, weperform all possible cube identification operations to the resulting complex. Thisfinal resulting complex is then isomorphic to Θ. Claim 4.4.
If a cube attachment operation can be performed on Σ to obtain anew complex Θ, then a sequence of cube attachment, fold and cube identificationoperations can be performed on Σ to obtain Θ.Suppose first that such a cube attachment operation attaches a d -cube to Σ with d = 2. It follows that there are edges e and f in Σ, each incident to a commonvertex and with labels s i and s j such that s i and s j are adjacent vertices of Γ. In Σ ON-QUASICONVEX SUBGROUPS 9 we see corresponding generalized edges e and f , each containing a common vertexand labeled by A i and A j respectively.Let x be an edge of ∆ incident to a vertex u ∈ A i and a vertex v ∈ A j . Thereis an edge in e with label u and an edge in f with label v . We perform a squareattachment operation to Σ by attaching a square with boundary label uvuv to Σalong these two edges. We perform such a square attachment for every edge x between A i and A j , and we denote the resulting complex by Π.Next, we perform all possible fold operations between pairs of edges that wereattached to Σ, and all possible cube identifications to the resulting complex. Let Π ′ denote the final resulting complex from these operations. Note that Σ is naturallya subcomplex of both Π and Π ′ .We claim that Π ′ is isomorphic to Θ. To prove this, it is enough to show that allvertices of Π \ Σ are identified to a common vertex of Π ′ . Thus, suppose that z and z ′ are vertices of Π \ Σ which are each contained in a square that was attached to Σ.The labels of these squares are abab and a ′ b ′ a ′ b ′ for some a, a ′ ∈ A i and b, b ′ ∈ A j .As the subgraph of ∆ induced by A i ∪ A i is connected (by the definition of Γ-partitegraphs), there is a path a = a, b = b, a , b . . . , a m − , b m − , a m = a ′ , b m = b ′ in ∆with a l ∈ A i and b l ∈ B j for all 1 ≤ l ≤ m . Correspondingly, there are squares withlabels a l b l a l b l and b l a l +1 b l a l +1 in Π for all 1 ≤ l ≤ m −
1. After folding, all verticescontained in one of these squares and which are not contained in Σ are identified.Thus, Π ′ is isomorphic to Θ.Suppose now that a cube attachment operation attaches a d -cube to Σ with d >
2. From the previous case, it readily follows that there is a sequence of squareattachment and fold operations that can be performed to Σ to produce a com-plex whose 1-skeleton is isomorphic to that of Θ. It is clear that we can obtain Θby performing a series of cube attachments and cube identifications to this complex.We are now ready to prove the lemma. Let Ω be a completion of Σ obtainedas the direct limit of a sequence Ω = Σ → Ω → Ω → . . . of complexes as inthe definition of a completion. By the two claims above, there exists a sequenceΠ = Σ → Π → Π → . . . of complexes such that Π i +1 is obtained from Π i byeither a cube attachment, cube identification or fold operation, and such that thereexists a subsequence Π = Π i , Π i , Π i , . . . such that Π i j = Ω j . Let Π be the directlimit of Π = Σ → Π → Π → . . . . It follows from the definition of a direct limitand the definition of the generalization of a complex that Π is isomorphic to Ω.In order to conclude that Π = Ω is indeed a completion of Σ, we now show thatit is also folded and cube-full. As Ω is a completion, it is folded and cube-full. Let φ : Ω → Ω be the collapsing map. To see that Ω is folded, let e and e ′ be edges ofΩ with the same label. As Ω is folded and as φ sends pairs of edges with the samelabel to pairs with the same label, we have that φ ( e ) = φ ( e ′ ). Thus, e and e ′ lie ina common generalized edge of Ω. Now, as e and e ′ have the same label and lie ina common generalized edge, we conclude that e = e ′ . Since no cube identificationoperations can be performed on Ω by definition, it follows that Ω is folded.To see that Ω is cube-full, consider a set e , . . . , e d of edges in Ω with d ≥
2, allincident to a common vertex and with labels t , . . . , t d corresponding to a d -cliqueof ∆. For 1 ≤ i ≤ d , let A i be the set in the decomposition of ∆ that contains thevertex t i . As there are no edges between vertices of A i for any i , it follows that A i = A j for all 1 ≤ i = j ≤ d . Thus, by the structure of Γ-partite graphs, the labels of φ ( e ) , . . . , φ ( e d ) span a d -clique in Γ. As Ω is cube-full, φ ( e ) , . . . , φ ( e d )are contained in a common d -cube. Consequently, as Ω is the generalization of Ω,there is a d -cube in Ω containing e , . . . , e d . Thus, Ω is cube-full. (cid:3) Let H be a finitely generated subgroup of W Γ generated by a finite set T of wordsin W Γ , and let R be the rose graph associated to T . Let R be the generalizationof R with respect to ∆. Let l , . . . , l m be a set of loops in R , based at the basevertex, which generate π ( R ), and let T = { w , . . . , w m } be the set of labels ofthese loops. We define the generalization H of H with respect to ∆ and T to bethe finitely generated subgroup of W ∆ that is generated by T . We remark that theisomorphism class of H very much depends on T and ∆, but not on T . When thereis no confusion regarding T and ∆, we simply say that H is the generalization of H . We will always use the bar notation as such to denote the generalization of agroup. Theorem 4.5.
Let Γ be a simplicial graph, and let ∆ be a Γ -partite graph. Let H be a finitely generated subgroup of W Γ generated by a finite set T of words in W Γ ,and let H < W ∆ be its generalization with respect to ∆ and T . Then (1) H is quasiconvex in W Γ if and only if H is quasiconvex in W ∆ , wherequasiconvexity is with respect to the standard generating sets of W Γ and W ∆ respectively. (2) If H is torsion-free, then so is H . (3) If H is not normal in W Γ , then H is not normal in W ∆ . (4) Suppose additionally that W Γ does not split as a product with a finite factor(equivalently, there does not exist a vertex of V (Γ) which is adjacent toevery other vertex of V (Γ) ). Then H is a finite index subgroup of W Γ ifand only if H is a finite index subgroup of W ∆ .Additionally, if ∆ does not contain induced -cycles (resp. -cycles), then W ∆ ishyperbolic (resp. -dimensional).Proof. The claims regarding W ∆ being hyperbolic or 2-dimensional follow from atheorem of Moussong [Mou88] and by definition respectively.Let R be the rose graph associated to T . Let R be the generalization of R , andlet T be the labels of a finite set of generators of π ( R, B ) where B is the basevertex. By definition, the generalization H is generated by T . Let R ′ be the rosegraph associated to T . It readily follows that R can be obtained from R ′ by asequence of fold operations. In particular, a completion of R is a completion of H .Let Ω be a completion of R . By Lemma 4.1, Ω is a completion of R and is there-fore a completion of H . As Ω is infinite if and only if Ω is infinite, (1) immediatelyfollows from Theorem 2.1(1).By Theorem 2.1(2), if H has torsion, then there is a loop in Ω whose label isa reduced word w with letters in a clique of ∆. It follows from the definition of aΓ-partite graph that w uses at most one letter from each set in the decompositionof ∆. Applying the collapsing map, one obtains a loop in Ω with label a word w ′ using letters in a clique of Γ, with each letter being used at most once. It followsthat w ′ is reduced. Again by Theorem 2.1(2), this implies that H has a torsionelement. Thus, (2) follows.We define a map φ : V (∆) → V (Γ) by, for each 1 ≤ i ≤ n and each a ∈ A i ,setting φ ( a ) = s i . Such a map naturally extends to a map from words in V (∆) towords in V (Γ). Moreover, if two generators s, t ∈ V (∆) of W ∆ commute, then it ON-QUASICONVEX SUBGROUPS 11 follows from the definition of a Γ-partite graph that φ ( s ) commutes with φ ( t ) in W Γ . Thus, φ defines a homomorphism φ : W ∆ → W Γ which is readily seen to besurjective.To see (3), suppose that H is not normal in W Γ . Let h be a word representingan element of H and g be a word representing an element of W Γ so that hgh − doesnot represent an element of H . By Theorem 2.1(4) and as Ω is a generalization ofΩ, there exists a word h , representing an element of H , so that φ ( h ) = h . As φ issurjective, there is a word g representing an element of W ∆ so that φ ( g ) = g . Let w be a reduced word equal to ghg − in W ∆ . Suppose, for a contradiction, that w represents an element of H . By Theorem 2.1(4), we have that there is a loop in Ωwith label w which is based at the base vertex. By Lemma 4.1, there is a loop withlabel φ ( w ) in Ω based at the base vertex. However, again by Theorem 2.1(4), thisimplies that φ ( w ) ∈ H . This is a contradiction as φ ( w ) is equal to ghg − in W Γ .Thus, ghg − does not represent an element of H and so H is not normal in W ∆ .Claim (4) follows from Theorem 2.1(5). (cid:3) The next corollary follows directly from Theorem 4.5 and Theorem 3.1.
Corollary 4.6.
Let H be a finitely generated non-quasiconvex subgroup of a (pos-sibly non-hyperbolic) RACG W Γ . Then there exists an infinite set { ∆ i | i ∈ N } of Γ -partite graphs such that, for each i , W ∆ i is -dimensional, hyperbolic and con-tains the generalization H of H (with respect to ∆ i and any finite generating setfor H ) as a non-quasiconvex subgroup. We finish this section by showing how Theorem 4.5 gives easy examples of non-quasiconvex subgroups of hyperbolic groups.
Example 4.7.
Let Γ be a complete bipartite graph. More specifically, let Γ be thegraph with vertex set { s , . . . , s k } , with k ≥
2, and with edges between s i and s j for all odd i and even j .Let w be the word s s . . . s k , and let H be the cyclic subgroup of W Γ generatedby w . The subgroup H can be seen to not be quasiconvex in W Γ by noting that W Γ = W { s ,s ,...,s k − } × W { s ,s ,...,s k } and that w n is equal, in W Γ , to the word( s s . . . s k − ) n ( s s . . . s k ) n . Alternatively, one can see this by constructing thecompletion of H and noting that it is an infinite cylinder that is tiled by squares.By Theorem 3.1, there exists a Γ-partite graph ∆ such that W ∆ is 2-dimensionaland hyperbolic. Then, by Theorem 4.5, the generalization H with respect to ∆and { w } is a non-quasiconvex subgroup of W ∆ . Additionally, as H is not normalin W Γ , Theorem 4.5 implies that H is not normal in W ∆ .The following example gives us Theorem B from the introduction. Example 4.8.
Let Γ be a 4-cycle with cyclically ordered vertices { s , . . . , s } . Let w = s s , w ′ = s s and T = { w, w ′ } . Note that w and w ′ are each of order twoin W Γ . The infinite dihedral subgroup H < W Γ generated by T is easily seen tobe non-quasiconvex in W Γ . The rose graph associated to T and the 1-skeleton of acompletion for it is shown in Figure 4.8.By Theorem 3.1, there exists a Γ-partite graph ∆ with decomposition A , . . . , A such that W ∆ is hyperbolic and 2-dimensional (such as the one in Figure 3). Thus,by Theorem 4.5, the generalization H of H is a non-quasiconvex subgroup of W ∆ .Moreover, by definition H is generated by the labels of all simple loops in R based atthe base vertex, where R is the rose graph associated to T . Thus, H is generated by s s s s . . . . . . Figure 4.1.
On the left is the rose graph R of Example 4.8, andon the right is the infinite 1-skeleton of a completion Ω of R . Thegraph R (resp. Ω) is obtained by replacing edges of R (resp. Ω)by generalized edges.all words in V (∆) of the from aa ′ with either { a, a ′ } ⊂ A ∪ A or { a, a ′ } ⊂ A ∪ A .In particular, all generators of H are of length 2. Example 4.9.
Given a join graph Γ = Γ ⋆ Γ , with Γ and Γ not cliques, theRACG W Γ = W Γ × W Γ contains many non-quasiconvex subgroups that can beused to produce non-quasiconvex subgroups of hyperbolic groups.Similarly, let Γ be a 4-cycle, and for integers i >
0, let Γ i be the suspensionof Γ i − . For i ≥ W Γ i is virtually isomorphic to Z i +2 and also contains manynon-quasiconvex subgroups.In the following example, W Γ is not a product. More complicated examples thatare not products can be similarly constructed. Example 4.10.
Let Γ be a graph consisting of a 6-cycle with cyclically orderedvertices s , . . . , s and an edge between s and s . Let T = { w = s s s s , w ′ = s s s s } , and let H < W Γ be generated by T . Let S and S be the 4-cycles ofΓ induced by { s , s , s , s } and { s , s , s , s } respectively. The group W Γ is theamalgamation of W S and W S over a finite group. As h w i is not quasiconvex in W S , it readily follows that H not quasiconvex in W Γ . (This can also easily be seenusing completions.) Thus, given a Γ-partite graph ∆ such that W ∆ is hyperbolic,we have that the generalization H of H is not quasiconvex in W ∆ .5. Non-quasiconvex free subgroups
In this section, we prove Theorem C from the introduction and give examples ofnon-locally quansiconvex hyperbolic groups that are fundamental groups of squarecomplexes with nonpositive sectional curvature.Theorem C follows immediately from Theorem 3.1, Lemma 3.3 and the following:
Theorem 5.1.
As in Example 4.7, let
Γ = { s , s , . . . , s k − } ⋆ { s , s , . . . , s k } be the complete bipartite graph with k ≥ and let H < W Γ be the cyclic subgroupgenerated by s s . . . s k . Let ∆ be a Γ -partite graph with path connectors andwith | A i | ≥ for each i . Then the generalization H is a finitely generated, non-quasiconvex, non-normal, free subgroup of W ∆ . Moreover, when k > , then H ∩ K is not normal in K for any finite index subgroup K < W ∆ .Proof. As in Example 4.7, let Γ be the complete bipartite graph { s , s , . . . , s k − } ⋆ { s , s , . . . , s k } with k ≥
2. Let A , . . . , A k be the corresponding decompositionof ∆. Let R denote a 2 k -cycle with label s s . . . s k . Recall that by definition, H ON-QUASICONVEX SUBGROUPS 13 is the subgroup of W ∆ generated by the labels of a set of generators of π ( R, B )where B is the base vertex. Note also that π ( R, B ) is free of some rank l .There exists a completion Ω of R which consists of a tiling of R × S by squares.(See Figure 2 of [DL19] for the case that k = 2.) By Lemma 4.1, the generalizationΩ of Ω is a completion for H . It readily follows from the description of Ω andTheorems 4.5 and 2.1 that H is non-quasiconvex, non-normal and torsion-free. Itremains to show that H is a free group. Since H is torsion-free, by Theorem 2.1(3)we see that H ∼ = π (Ω). We now show that π (Ω) is a free group of rank l .We first introduce some new complexes. For 1 ≤ i ≤ k , let R i denote thesubgraph of R consisting of the generalized edges with labels A i and A i +1 (takenmodulo 2 k ), and let v i denote the vertex incident to both these generalized edges.Choose one edge in each of these two generalized edges to form a maximal tree,leading to a choice of basis loops for π ( R i ). Let H i be the subgroup of W ∆ generated by the labels of these loops. By Theorem 3.17 from [DL20], H i is a freegroup with this generating set as a free basis. Let Ω i denote the generalizationof a single 2-cube with s i s i +1 s i s i +1 as its boundary label. It can be seen thatΩ i is a completion for H i . Moreover, since H i is free, it is torsion-free, and so π (Ω i ) ∼ = H i ∼ = π ( R i ).It is evident from the description of Ω and the definition of a generalization ofa complex, that Ω is a direct limit of a sequence of complexes Σ i , where Σ = R and Σ j +1 is obtained from Σ j by gluing a copy of Ω i for some i to Σ j along a copyof R i in each of these. Now for j ≥
0, van Kampen’s theorem says that π (Σ j +1 )is the pushout of the diagram π (Σ j ) ← π ( R i ) → π (Ω i ). From what we provedabove, the map π ( R i ) → π (Ω i ) is an isomorphism, and so its inverse provides anatural map π (Ω i ) → π (Σ j ). By the universal property of pushouts, this map,together with the identity map of π (Σ j ) induces a unique map φ : π (Σ j +1 ) → π (Σ j ). Moreover, the natural map π (Σ j ) → π (Σ j +1 ) is an inverse for φ . Thus, π (Σ j ) ∼ = π (Σ j +1 ). This, together with the fact that π (Σ ) is a free group of rank l completes the proof that H is free of rank l .To see the last claim, choose distinct a, a ′ ∈ A and b, b ′ ∈ A . As there are noedges between two vertices of A i and as s is not adjacent to s in Γ, it followsby Tits’ solution to the word problem (see [Dav15]) that for every p, q ∈ Z \ { } ,( aa ′ ) p and ( bb ′ ) q ( aa ′ ) p ( b ′ b ) q are reduced words. In Ω, the base vertex is containedin four distinct edges, labeled by s , s , s k and s k − . Thus, by Theorem 2.1(4)and the structure of Ω, we have that, for every p, q ∈ Z \ { } , ( aa ′ ) p representsan element of H and ( bb ′ ) q ( aa ′ ) p ( b ′ b ) q does not represent an element of H . Now,given any finite index subgroup K < W ∆ , there exists some p, q ∈ Z \ { } such that( bb ′ ) q ∈ K and ( aa ′ ) p ∈ K ∩ H . The claim now follows. (cid:3) The next lemma provides a construction of Γ-partite graphs with nonpositivesectional curvature.
Lemma 5.2.
Let Γ be a -cycle, and let ∆ be a Γ -partite graph with path connectors.Then ∆ has nonpositive sectional curvature.Proof. Let s , . . . , s be the labels of a cyclic ordering of the vertices of Γ, andlet A , . . . , A be the corresponding decomposition of ∆. Let Θ be a connected,spurless subgraph of ∆ which contains at least one edge. Let V i be the number ofvertices of Θ contained in A i , and let E i be the number of edges of Θ connecting a vertex of A i to a vertex of A i +1 (taken modulo 4). We need to check that2 − P i =1 V i + P i =1 E i is at most 0.As the edges in ∆ between A i and A i +1 form a path, we have that E i ≤ V i − V i = V i +1 = 0 and E i ≤ V i , V i +1 ) whenever V i = V i +1 .Suppose first that V i = 0 for all i . We then get: κ (Θ) ≤ − X i =1 V i + X i =1 ǫ i ( V i −
12 ) + X i =1 (1 − ǫ i ) min( V i , V i +1 )where ǫ i = 1 if | V i | = | V i +1 | and ǫ i = 0 otherwise. Regrouping terms and notingthat ǫ i ( V i − min( V i , V i +1 )) = 0 for all i , we get: κ (Θ) ≤ − X i =1 V i − X i =1 ǫ i + X i =1 ǫ i ( V i − min( V i , V i +1 )) + X i =1 min( V i , V i +1 )= 2 − A − B, where A = X i =1 V i − X i =1 min( V i , V i +1 ) and B = X i =1 ǫ i . We prove below that A + B ≥
2. From this, it follows that κ (Θ) ≤ − A − B ≤ M and m respectively be the maximum and minimum of { V i | ≤ i ≤ } . If M = m , then we necessarily have that ǫ i = 1 for all i . It follows that B = 2. Thus, A + B ≥
2, since A ≥
0. Now suppose
M > m . Without loss of generality, assumethat V = m . Then A = m + V + V + V − ( m + min( V , V ) + min( V , V ) + m ) . We have V i = M for some i ∈ { , , } . In each of these cases, one readily verifiesthat A ≥ M − m , (using the observation that V i − min( V i , V i ± ) ≥ i ).From this, it follows that if M − m ≥
2, then A + B ≥ M − m + 0 ≥
2. Finally,suppose M − m = 1, so that A ≥
1. Observe that if ǫ i = 1 for some i , then thereexists j = i such that ǫ j = 1. In this case, B ≥
1, and once again A + B ≥
2. In thelast remaining case, namely when M − m = 1 and ǫ i = 0 for all i , we necessarilyhave V = V = m + 1 and V = m , and it is easily verified that A ≥
2. Therefore, A ≥
2. We have thus verified that the sectional curvature is at most 0 when V i = 0for all i .If V i = V j = 0 for some i = j , then Θ necessarily contains a spur. This followsas Θ is connected, contains at least one edge and the edges in ∆ between A i and A i +1 form a simple path. Thus, it cannot be that V i = V j = 0 for some i = j .The only remaining case to check, up to relabeling, is that where V = 0 and V , V and V are non-zero. Suppose this is the case. Each vertex of A (respectivelyof A ), is adjacent to at most two vertices of A . As each edge of Θ is incident toa vertex of A ∪ A , we see that the total number of edges of Θ must be at most2( V + V ). We thus get: κ (Θ) ≤ − X i =1 V i + ( V + V )If V >
1, this implies that κ (Θ) ≤ V = 1, for then one of the vertices of V must be a spur. We have thusestablished the claim. (cid:3) ON-QUASICONVEX SUBGROUPS 15
By applying Theorem 3.1 and the following theorem, we obtain explicit examplesof non-locally quasiconvex, hyperbolic groups that are fundamental groups of right-angled square complexes with nonpositive sectional curvature.
Theorem 5.3.
Let Γ be a -cycle, and let ∆ be a Γ -partite graph with path con-nectors, no -cycles and no -cycles. Then W ∆ has a finite-index, torsion-free, hy-perbolic subgroup K that is the fundamental group of a right-angled square complexwith nonpositive sectional curvature. Moreover, K contains a finitely generated,non-quasiconvex, free subgroup. In particular, K is not locally quasiconvex.Proof. Let K be the commutator subgroup of W ∆ . Then K is hyperbolic, as it is afinite-index subgroup of the hyperbolic group W ∆ . Morever, K is the fundamentalgroup of a right-angled square complex with nonpositive sectional curvature as ∆has nonpositive sectional curvature by Lemma 5.2.Let H be the finitely generated, non-quasiconvex free subgroup of W ∆ given byTheorem 5.1. It follows that H ∩ K is a non-quasiconvex, free subgroup of K . (cid:3) Not finitely presentable, non-quasiconvex subgroups
In this section we show how Γ-partite graphs with cycle connectors provideexamples of finitely generated, non-quasiconvex subgroups of hyperbolic RACGswhich contain closed surface subgroups (Theorem 6.2). We also give, using ourmethod, a construction of finitely generated, not finitely presentable subgroups ofhyperbolic RACGs (Theorem 6.3).
Lemma 6.1.
Let C be a cycle with cyclically-ordered vertices p , p , . . . , p k andwith k ≥ . Let A = { p p i | i is odd , i > } and B = { p p i | i is even , i > } Then the subgroup
K < W C generated by A ∪ B is isomorphic to the fundamentalgroup of a closed hyperbolic surface.Proof. By applying Theorem 2.1 to a finite completion associated to K , we concludethat K is a torsion-free, finite-index subgroup of W C . The group W C is well-knownto act geometrically by reflections on the hyperbolic plane (see [Dav15] for instance).Thus, K is isomorphic to the fundamental group of a closed hyperbolic surface. (cid:3) Theorem 6.2.
Let Γ be a simplicial graph, ∆ be a Γ -partite graph with cycle con-nectors, and let H be a non-quasiconvex subgroup of W Γ . Then H is not quasiconvexin W ∆ and contains a closed surface subgroup.Proof. Let Ω be a completion of H . The generalization Ω is a completion of H by Lemma 4.1. As H is not quasiconvex in W Γ , Theorem 2.1 implies that Ω isinfinite. In particular, some cube attachment operation must have been performedto construct Ω, and it follows that Ω contains a square with label s i s j s i s j where s i and s j are adjacent vertices of Γ. We see a corresponding generalized square c (whose image under the collapsing map is c ) in Ω with label A i A j A i A j .Let v be a vertex of Ω that is incident to both a generalized edge e labeled by A i and a generalized edge f labeled by A j . Let K be the subgroup of W ∆ generatedby the labels of all simple loops based at v and contained in e ∪ f . By Lemma 6.1and as a subgroup of W ∆ generated by V ( C ) is isomorphic to W C for any inducedsubgraph C ⊂ ∆ (see [Dav15]), K is a closed hyperbolic surface subgroup. Let w be the label of a path in Ω from the base vertex to v . By Theorem 2.1(4), H contains a w conjugate of K . (cid:3) Theorem 6.3.
Let Γ be a -cycle, and let ∆ be a Γ -partite graph with cycleconnectors, no -cycles and no -cycles. Then W ∆ is -dimensional, hyperbolicand contains a finitely generated, not finitely presentable (and in particular, non-quasiconvex) subgroup.Proof. We immediately have that W ∆ is 2-dimensional and hyperbolic as ∆ has no3-cycles and no 4-cycles. Let the vertices of Γ be cyclically labeled s , . . . , s . Let H be the subgroup of W Γ generated by the word s s s s .Let H be the generalization of H . Let K be the commutator subgroup of W ∆ ,and let H ′ := H ∩ K . As W ∆ is 2-dimensional, K has cohomological dimensionat most 2. We will show that H ′ is a normal, infinite-index, infinite, non-freesubgroup of K . It will then follow from a well-known theorem of Bieri [Bie81] (seealso [Wis19, Theorem 9.1]) that H ′ is not finitely presentable.As H is torsion free and has infinite index in W Γ , we have by Theorem 4.5 that H has infinite index in W ∆ and is torsion-free. In particular, H ′ is infinite andhas infinite index in K . By Theorem 6.2, H contains a closed surface subgroup S .As K has finite index in W ∆ , it follows that H ′ = H ∩ K contains a finite-indexsubgroup of S . As any finite-index subgroup of S is also isomorphic to a closedsurface group, it follows that H ′ is not free. It remains to show that H ′ is normalin K .Let Ω be the completion of H . The generalization Ω of Ω is a completion of H byLemma 4.1. The 1-skeleton of the complex Ω is isomorphic to an infinite bipartitegraph (see Figure 2 of [DL19]). Moreover, it is readily seen that given two vertices u and v (and corresponding vertices u and v in Ω) in a common component of thisbipartite graph, there is a label preserving automorphism of Ω (resp. Ω) sending u to v (resp. u to v ).To see that H ′ is normal in K , let k be a reduced word representing an elementof K and h be a reduced word representing an element of H ′ . It is enough toshow that khk − ∈ H . Let γ be a path in Ω labeled by khk − and based at thebase vertex. Such a path exists, since for every s ∈ V (∆) and for every vertex v of Ω, there is an edge labeled by s incident to v . As the length of k is even (as k represents an element of the commutator subgroup K ), as there is a loop in Ω basedat the base vertex with label h (by Theorem 2.1(4)) and as Ω is folded, it followsfrom the previous paragraph that γ is a loop in Ω. Thus, again by Theorem 2.1(4), khk − represent an element of H . We have thus shown that H ′ is normal in K ,concluding our proof. (cid:3) References [ABC +
91] J. M. Alonso, T. Brady, D. Cooper, V. Ferlini, M. Lustig, M. Mihalik, M. Shapiro, andH. Short,
Notes on word hyperbolic groups , Group theory from a geometrical viewpoint(Trieste, 1990), World Sci. Publ., River Edge, NJ, 1991, Edited by Short, pp. 3–63.MR 1170363[BB97] Mladen Bestvina and Noel Brady,
Morse theory and finiteness properties of groups ,Invent. Math. (1997), no. 3, 445–470. MR 1465330[BBD07] Josh Barnard, Noel Brady, and Pallavi Dani,
Super-exponential distortion of subgroupsof CAT( − ) groups , Algebr. Geom. Topol. (2007), 301–308. MR 2308946 ON-QUASICONVEX SUBGROUPS 17 [BDR13] Noel Brady, Will Dison, and Timothy Riley,
Hyperbolic hydra , Groups Geom. Dyn. (2013), no. 4, 961–976. MR 3134032[Bie81] Robert Bieri, Homological dimension of discrete groups , second ed., Queen Mary Col-lege Mathematical Notes, Queen Mary College, Department of Pure Mathematics,London, 1981. MR 715779[Bra99] Noel Brady,
Branched coverings of cubical complexes and subgroups of hyperbolicgroups , J. London Math. Soc. (2) (1999), no. 2, 461–480. MR 1724853[Dav15] Michael W. Davis, The geometry and topology of Coxeter groups , Introduction tomodern mathematics, Adv. Lect. Math. (ALM), vol. 33, Int. Press, Somerville, MA,2015, pp. 129–142.[DL19] Pallavi Dani and Ivan Levcovitz,
A study of subgroups of right-angled Coxeter groupsvia Stallings-like techniques , arXiv:1908.09046, 2019.[DL20] ,
Right-angled Artin subgroups of right-angled Coxeter and Artin groups ,arXiv:2003.05531, 2020.[JNW19] Kasia Jankiewicz, Sergey Norin, and Daniel Wise,
Virtually fibering right-angled cox-eter groups , J. Inst. Math. Jussieu (2019), Published online.[Kap99] Ilya Kapovich,
A non-quasiconvexity embedding theorem for hyperbolic groups , Math.Proc. Cambridge Philos. Soc. (1999), no. 3, 461–486. MR 1713122[Mit98] M. Mitra,
Cannon–Thurston maps for trees of hyperbolic metric spaces , J. DifferentialGeometry (1998), 135–164.[Mou88] Gabor Moussong, Hyperbolic Coxeter groups , ProQuest LLC, Ann Arbor, MI, 1988,Thesis (Ph.D.)–The Ohio State University. MR 2636665[Rip82] E. Rips,
Subgroups of small cancellation groups , Bull. London Math. Soc. (1982),no. 1, 45–47. MR 642423[Thu97] William P. Thurston, Three-dimensional geometry and topology. Vol. 1 , PrincetonMathematical Series, vol. 35, Princeton University Press, Princeton, NJ, 1997, Editedby Silvio Levy. MR 1435975[Wis04] D. T. Wise,