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Publicly Available Published by De Gruyter July 6, 2017

Subgroups of relatively hyperbolic groups of Bredon cohomological dimension 2

  • Eduardo Martínez-Pedroza EMAIL logo
From the journal Journal of Group Theory

Abstract

A remarkable result of Gersten states that the class of hyperbolic groups of cohomological dimension 2 is closed under taking finitely presented (or more generally FP2) subgroups. We prove the analogous result for relatively hyperbolic groups of Bredon cohomological dimension 2 with respect to the family of parabolic subgroups. A class of groups where our result applies consists of C(1/6) small cancellation products. The proof relies on an algebraic approach to relative homological Dehn functions, and a characterization of relative hyperbolicity in the framework of finiteness properties over Bredon modules and homological Isoperimetric inequalities.

1 Introduction

It is a remarkable result of Gersten that the class of hyperbolic groups of cohomological dimension at most 2 is closed under taking finitely presented subgroups, see Theorem 1.1. The dimension assumption is sharp since Brady exhibited a hyperbolic group of cohomological dimension 3 containing a finitely presented subgroup which is not hyperbolic [5].

Theorem 1.1 (Gersten [14, Theorem 5.4]).

Let G be a hyperbolic group and suppose cd(G)2. If H is a subgroup of type FP2, then H is hyperbolic.

In this article we show that such subgroup phenomena hold in the class of relatively hyperbolic groups of Bredon cohomological dimension 2 with respect to parabolic subgroups. We use the approach to relative hyperbolicity by Bowditch [4] which is equivalent to the approach by Osin [27] for finitely generated groups [27, Theorem 6.10].

Definition 1.2 (Bowditch relatively hyperbolic groups [4]).

A graph Γ is fine if every edge is contained in only finitely many circuits (embedded closed paths) of length n for any integer n. A finitely generated group G is hyperbolic relative to a finite collection of subgroups 𝒫 if G acts on a connected, fine, hyperbolic graph Γ with finite edge stabilizers, finitely many orbits of edges, and 𝒫 is a set of representatives of distinct conjugacy classes of vertex stabilizers (such that each infinite stabilizer is represented). Subgroups of G which are conjugate to a subgroup of some P𝒫 are called parabolic subgroups.

A family of subgroups of a group G is a non-empty collection of subgroups closed under conjugation and subgroups. The theory of (right) modules over the orbit category 𝒪(G) was established by Bredon [8], tom Dieck [29] and Lück [20]. In the case that is the trivial family, the 𝒪(G)-modules are G-modules. In this context, the notions of cohomological dimension 𝖼𝖽(G) and finiteness properties FPn, for the pair (G,) are defined analogously as their counterparts 𝖼𝖽(G) and FPn, see Section 3.

To simplify the notation in our statements we introduce the following non-standard terminology. A group G is hyperbolic relative to a family of subgroups if there is a finite collection of subgroups 𝒫 generating the family such that G is hyperbolic relative to 𝒫. If HG and is a family of subgroups, we denote by H the family of subgroups {KH:K}. Our main result is the following.

Theorem 1.3.

Let G be hyperbolic relative to a family F containing all finite subgroups. If cdF(G)2 and HG is of type FP2,FH, then H is hyperbolic relative to FH.

Given a group G and a family of subgroups , a model for EG is a G-complex X such that all the isotropy groups of X belong to , and for every Q the fixed-point set XQ is a non-empty contractible subcomplex of X. The geometric dimension 𝗀𝖽(G) is defined as the smallest dimension of a model for EG. It is known that

𝖼𝖽(G)𝗀𝖽(G)max{3,𝖼𝖽(G)},

see the work of Lück and Meintrup [21, Theorem 0.1].

Examples of Brady, Leary and Nucinkis [6], and Leary and Fluch [12] show that the equality of 𝖼𝖽(G) and 𝗀𝖽(G) fails for some non-trivial families. For the case that is the trivial family, whether this equality holds is the outstanding Eilenberg-Ganea conjecture. The use of cohomological dimension instead of geometric dimension in Theorem 1.3 is justified by the unknown Eilenberg–Ganea phenomena for general families.

In practice, we expect most applications of Theorem 1.3 to be in the geometric case 𝗀𝖽(G)2 which lies outside the abstract framework of Bredon modules.

1.1 A version of Theorem 1.3 outside the framework of Bredon modules

By a toral relatively hyperbolic group we mean a finitely generated torsion-free group G which is hyperbolic relative to a finite collection 𝒫 of abelian subgroups. By [27, Theorem 1.1], all subgroups in 𝒫 are finitely generated.

Theorem 1.4.

Let G be toral relatively hyperbolic group, let F be the family of parabolic subgroups, and suppose gdF(G)2. If HG is finitely presented with finitely many conjugacy classes of maximal parabolic subgroups, then H is toral relatively hyperbolic.

Theorem 1.4 follows from 𝖼𝖽(G)𝗀𝖽(G) and Proposition 1.5 below. Recall that a collection of subgroups 𝒫 of a group G is malnormal (almost malnormal) if for any P,P𝒫 and gG, either g-1PgP is trivial (respectively, finite), or P=P and gP.

Proposition 1.5 (Proposition 3.10).

If G is finitely presented and F is a family generated by a malnormal finite collection of finitely generated subgroups, then G is FP2,F.

Proof of Theorem 1.4.

Since G is torsion-free, any two maximal parabolic subgroups of G have either trivial intersection or they are equal [25, Lemma 2.2]. Hence the same statement holds for the maximal parabolic subgroups of H. Since all parabolic subgroups of H are finitely generated, Proposition 1.5 implies that H is FP2,H. Then Theorem 1.3 implies that H is toral relatively hyperbolic. ∎

We raise the following question on the hypothesis of Theorem 1.4, remark on a simple example that might shed some light on the situation, and quote a related result.

Question 1.6.

Let G be a toral relatively hyperbolic group, let be the family of parabolic subgroups, and suppose 𝖼𝖽(G)2. Suppose that H is finitely presented. Is H toral relatively hyperbolic?

Remark 1.7.

There is a toral relatively hyperbolic G with 𝖼𝖽(G)2 where is the family of parabolic subgroups, and such that G contains a free group of finite rank with infinitely many conjugacy classes of maximal parabolic subgroups. In particular, H is not hyperbolic relative to H, the group H is not FP2,H as a consequence of [28, Proposition 3.6.1], but H is hyperbolic and hence toral relatively hyperbolic.

Indeed, let H be a free group of rank 2, and let G be a hyperbolic free by cyclic group Hφ; for the existence of an automorphism φ𝖠𝗎𝗍(H) such that the resulting semidirect product is hyperbolic we refer the reader to [3]. Let g be a non-trivial element of H that is not a proper power, and let be the family generated by P=g. Then G is hyperbolic relative to by [4, Theorem 7.11]. Since H is normal in G, the subgroup H has non-trivial intersection with any conjugate of P. Hence H has infinitely many conjugacy classes of maximal parabolic subgroups, and in particular is not hyperbolic relative to H. It is left to remark that 𝗀𝖽(G)2. Since H has rank two, G is the fundamental group of a finite 2-dimensional locally 00 complex X, see [7]. Let γ be a closed geodesic in X representing the conjugacy class of g, and let Y be the space obtained by taking the universal cover X~ of X and coning-off each geodesic of X~ covering γ. Observe that G acts on Y and that Y is a 2-dimensional model for EG.

A 11 version of Theorem 1.4 was known by Hanlon and the author [18, Theorem 5.15]. This result addresses Question 1.6 in a particular geometric setting. We quote this result in the framework of toral relatively hyperbolic groups, and remark that the statement in the cited reference is slightly more general.

Theorem 1.8 ([18, Theorem 5.15]).

Let G be a toral relatively hyperbolic group, let F be the family of parabolic subgroups, and suppose there is a cocompact 2-dimensional 11 model of EFG with trivial edge stabilizers. If HG is finitely presented, then H is toral relatively hyperbolic.

1.2 An application to small cancellation products

The studies on small cancellation groups by Greendlinger imply that a torsion-free C(1/6) group has cohomological dimension 2, see [11]. An analogous statement holds for all small cancellation products as defined in [22, Chapter V].

Theorem 1.9 (Theorem 6.3).

Let G be the quotient of a free product i=1mPi/R where R is a finite symmetrized set. If R is satisfies the C(1/6)-small cancellation hypothesis and G is torsion-free, then gdF(G)2 where F is the family generated by {Pi:1im}.

A group G as in the proposition is hyperbolic relative to {Pi}i=1m, see for instance [27, p. 4, (II)]. Theorems 1.3 and 1.9 yield Corollary 1.10 below. The statement of the corollary for finitely presented C(1/6)-small cancellation groups was known [14, Theorem 7.6].

Corollary 1.10.

Let G be a torsion-free quotient i=1mPi/R of a free product i=1mPi where R is a finite set satisfying the C(1/6)-small cancellation condition. Let F be the family generated by {Pi}i=1m. If HG is FP2,FH, then H is hyperbolic relative to FH.

1.3 A characterization of relative hyperbolicity

The proof of Theorem 1.3, relies on a characterization of relatively hyperbolic groups stated below.

The homological Dehn function of a space is a generalized isoperimetric function describing the minimal volume required to fill cellular 1-cycles with cellular 2-chains, see Section 4 for definitions. The homological Dehn function FVG: of a finitely presented group G is the homological Dehn function of the universal cover of a K(G,1) with finite 2-skeleton; the growth rate of this function is an invariant of the group [1, 16, 14]. Recall that a pair of functions f and g from to {} have equivalent growth rate if fg and gf where fg means that there exists a constant C with 0<C< such that f(n)Cg(Cn+C)+Cn+C for all n.

Gersten observed that one can define FVG only assuming that G is an FP2 group, and discovered the following characterization of word-hyperbolic groups.

Theorem 1.11 (Gersten [14, Theorem 5.2]).

A group G is hyperbolic if and only if G is FP2 and FVG(n) has linear growth.

Given a group G and a family of subgroups , under the assumption that G is FP2,, we define the homological Dehn function of G relative to which we denote by FVG,. The growth rate of FVG, is an invariant of the pair (G,), see Theorem 4.4. In the case that consists of only the trivial subgroup and G is finitely presented (or more generally FP2), the definition of FVG, coincides with the homological Dehn function FVG.

Theorem 1.12.

Let G be a group and let P be a finite collection of subgroups generating a family F.

  1. Suppose that 𝒫 is an almost malnormal collection. If G is FP2, and FVG,(n) has linear growth, then G is hyperbolic relative to 𝒫.

  2. Suppose that contains all finite subgroups of G. If G is hyperbolic relative to 𝒫, then G is F2, and FVG,(n) has linear growth.

This result could be interpreted as an algebraic version of [17, Proposition 2.50] or [23, Theorem 1.8]. The proof of Theorem 1.12 (1) is given in Section 6. Theorem 1.12 (2) is a result of Przytycki and the author [24, Corollary 1.5].

Remark 1.13.

The assumption that 𝒫 is an almost malnormal collection in Theorem 1.12 (1) cannot be removed from the statement. Let G=, let P=0, and let be the family of subgroups of P. The real line is a cocompact model for EG when considering the action on which 0 acts trivially and 0 acts by translations. In this case, G is FP2,, the function FVG, is trivial, and G is not hyperbolic relative to 𝒫.

1.4 The proof of Theorem 1.3

The main step in the proof of Theorem 1.3 is the following subgroup theorem for homological Dehn functions. The proof of Theorem 1.14 will be given in Section 5.2.

Theorem 1.14.

Let G be a group and F a family of subgroups. Suppose that G admits a cocompact EFG and cdF(G)2. If HG is of type FP2,FH, then FVH,FHFVG,F.

Remark 1.15.

In the case that is the trivial family, a version of this result is implicit in the work of Gersten [14]. Another version in the case that is the trivial family is [19, Theorem 1.1] which assumes that there is a 2-dimensional compact K(G,1) and H is finitely presented. Recall that finite presentability is not equivalent to FP2, see [2, Example 6(3)].

Theorem 1.16 ([24, Theorem 1.1]).

Let G be a hyperbolic group relative to a family F containing all finite subgroups. Then there is a cocompact model for EFG.

Proof of Theorem 1.3.

By Theorems 1.12 (2) and 1.16, the function FVG,(n) has linear growth, and that G admits a cocompact EG, respectively. Theorem 1.14 implies that FVH,HFVG,, and hence FVH,H has linear growth. Since H is FP0,H, the family H is generated by a finite collection 𝒬 of subgroups [28, Proposition 3.6.1]. Assume that the collection 𝒬 has minimal size; observe that no pair of subgroups of 𝒬 are conjugate in H. Since the collection of maximal parabolic subgroups of G is almost malnormal [25, Lemma 2.2], then 𝒬 is an almost malnormal collection of subgroups in H. By Theorem 1.12 (1), H is hyperbolic relative to 𝒬. ∎

1.5 Outline

The rest of the article is organized as follows. Section 2 contains preliminary material including a brief review of Bredon modules. Section 3 contains a combinatorial proof that FP1, is equivalent to F1,, an expected result for which we could not find a proof in the literature. Section 4 introduces the notion of homological Dehn function FVG, for a pair (G,) and discuss its algebraic and topological approaches. Section 5 contains the proof of Theorem 1.14. Section 6 discusses relative hyperbolicity and contains the proofs of Theorem 1.9 and Theorem 1.12.

2 Preliminaries

2.1 G-spaces and G-maps

In this article, all group actions are from the left, all spaces are combinatorial complexes and all maps between complexes are combinatorial. All group actions on complexes are by combinatorial maps. Moreover, all group actions are assumed to have no inversions; this means that if a cell is fixed setwise, then it is fixed pointwise. A G-complex X is cocompact if there are finitely many orbits of cells. For a G-complex X, the G-stabilizer of a cell σ is denoted by Gσ, and the fixed-point set of HG is denoted by XH. Since there are no inversions, the fixed-point set XH is a subcomplex of X. A path IX in a complex X is a combinatorial map from an oriented subdivided interval; in particular, a path has an initial vertex and a terminal vertex.

Definition 2.1 (Combinatorial complexes and maps [9, Chapter I Appendix]).

A map XY between CW-complexes is combinatorial if its restriction to each open cell of X is a homeomorphism onto an open cell of Y. A CW-complex X is combinatorial provided that the attaching map of each open cell is combinatorial for a suitable subdivision.

2.2 Modules over the orbit category

Let be a family of subgroups of G. The orbit category 𝒪(G) is the small category whose objects are the G-spaces of left cosets G/H with H and morphisms are the G-maps among them. The set of morphisms from G/K to G/L is denoted by [G/K,G/L].

The category of right 𝒪(G)-modules is the functor category of contravariant functors from 𝒪(G) into the category of abelian groups AB. More concretely, a right module over 𝒪(G) is a contravariant functor 𝒪(G)AB, and a morphism MN between right modules over 𝒪(G) is a natural transformation from M to N. For a 𝒪(G)-module M and H, we denote by M[G/H] the image of the object G/H by the functor M. The abelian group M[G/H] is called the component of M at G/H. For the rest of this article by an 𝒪(G)-module we shall mean a right 𝒪(G)-module.

Since the category of 𝒪(G)-modules is a functor category, several constructions of the category of abelian groups AB can be carried out in 𝒪(G)-modules by performing them componentwise. As a consequence it is a complete, cocomplete, abelian category with enough projectives. In particular, computing kernels, images, and intersections reduces to componentwise computations in AB. For instance, a sequence KLM of 𝒪(G)-modules is exact if and only if for every H the sequence K[G/H]L[G/H]M[G/H] of abelian groups is exact.

2.3 Free 𝒪(G)-modules

Let S be a (left) G-set. Consider the contravariant functor

𝒪(G)𝑆SET

that maps the object G/H to the fixed-point set SH and maps a morphism

G/H𝜑G/K

given by φ(H)=gK to the function

XKXH

given by xg.x. The 𝒪(G)-module induced by S, denoted by [,S], is the composed functor

𝒪(G)𝑠SETAB

where the second functor is the free abelian group construction X[X]. For a subgroup KG, the module induced by the G-set G/K is denoted by [,G/K].

The module [,G/G] is denoted by G,. We note that each component [G/K,G/G] is an infinite cyclic group with canonical generator the only element of G/G.

An 𝒪(G)-module F is free if F is isomorphic to [,S] for S a G-set with isotropy groups in . If, in addition, the G-set S has finitely many orbits, then [,S] is a finitely generated free 𝒪(G)-module.

2.4 Augmentation maps

Let S be a G-set and consider the 𝒪(G)-module [,S]. For K, the component [G/K,S] is by definition the free abelian group [SK] with SK as a free basis. The augmentation map of [,S] is the morphism ϵ:[,S]G, whose components ϵG/K:[SK] are defined by mapping each element of SK to the canonical generator of [G/K,G/G]. The main observation is that ϵ is surjective if and only if the fixed-point set SK is not empty for every K.

2.5 Schanuel’s lemma

The following proposition is a well-known result of homological algebra that is used several times through the article.

Proposition 2.2 (Schanuel’s lemma [10, Chapter VIII, 4.2]).

Let

0PnPn-1P0M0

and

0PnPn-1P0M0

be exact sequences of OF(G)-modules such that Pi and Pi are projective for in-1.

  1. If n is even, then

    P0P1P2P3PnP0P1P2P3Pn.
  2. If n is odd, then

    P0P1P2P3PnP0P1P2P3Pn.

2.6 The cellular 𝒪(G)-chain complex of a G-CW-complex

Let G be a group, let be a family of subgroups, and let X be a G-complex. These data induce a canonical contravariant functor

𝒪(G)TOP

that maps the object G/H to the fixed-point set

XH={xX:for all hH,h.x=x},

and maps a G-map

G/H𝜑G/K

given by φ(H)=gK to the cellular map

XKXH

given by xg.x. Observe that XH is a subcomplex of X since we are assuming that our actions are cellular and without inversions. The composition of the contravariant functor 𝒪(G)TOP with any covariant functor TOPAB (as a homological functor) is a right 𝒪(G)-module.

The augmented cellular 𝒪(G)-chain complex of X, which will be denoted by C*(X,𝒪(G)), is the contravariant functor

𝒪(G)TOPchain complexes,G/HXHC*(XH,)

where the second functor is the standard cellular augmented -chain complex functor.

Remark 2.3.

If X is a cocompact G-cell-complex such that XK is n-acyclic for every K and all the isotropy groups of X belong to , then the induced chain complex of 𝒪(G)-modules

Cn(X,𝒪(G))n2C1(X,𝒪(G))1C0(X,𝒪(G))ϵG,0,

is an exact sequence of finitely generated free 𝒪(G)-modules.

3 Finiteness properties

Let G be a group and let be a family of subgroups.

Definition 3.1.

The group G is of type Fn, if there is a cocompact G-complex X such that all the isotropy groups of X belong to , and for every K the fixed-point set XK is a non-empty and (n-1)-connected subcomplex of X. A complex X satisfying these conditions is called an an Fn,-complex for G. The group G is of type F if there is a cocompact model for E(G).

Definition 3.2.

The group G is said to be of type FPn, if there is a resolution of 𝒪(G)-modules

Pnn2P11P00

such that for 0in the module Pi is finitely generated and projective. Such a resolution is called an FPn,-resolution. It is equivalent to require that for all i with 0in, the module Pi is finitely generated and free, see for example [10, Chapter VIII, 4.3].

The main result of the section is the following proposition whose proof, as well as the proof of its corollary, are postponed to Subsection 3.1.

Proposition 3.3.

If G is FP1,F, then G is F1,F.

Definition 3.4.

The family generated by a collection of subgroups 𝒫 of G consists of all subgroups KG such that g-1KgP for some gG and P𝒫.

Corollary 3.5.

If G is FP1,F and F is a family generated by a finite and almost malnormal collection P, then there is an F1,F-graph with finite edge G-stabilizers.

The second part of the section will recall the definition of the coned-off Cayley complex of a relative presentation, and observe that these complexes provide F2,-complexes in some cases.

3.1 Proof of Proposition 3.3

Take an FP1,-resolution consisting of only free 𝒪(G)-modules, see for example [10, Chapter VIII, 4.3],

[,T]𝑝[,V]ϵ𝔾,0

where [,V]ϵG, is the augmentation map. Then T and V are G-sets with isotropy groups in and finitely many G-orbits. For each K the sequence of abelian groups arising by taking the G/K-component

[TK]𝑝[VK]ϵ0

is exact.

Step 1.

For each tT there is a finite directed Gt-graph Γt=(Vt,Et) such that Vt is a subset of V and

  1. Γt is connected when considered as an undirected graph,

  2. the Gt-action on Vt is the restriction of the G-action on V,

  3. the Gt-stabilizer of an edge eEt is the intersection of the Gt-stabilizers of its endpoints,

  4. eEt(e+-e-)=p(t).

Proof.

Since T has isotropy groups in , we have that Gt and hence the sequence

[TGt]𝑝[VGt]ϵ0

is exact. Therefore the fixed-point set VGt is non-empty. Choose tVGt. Consider the expression

p(t)=vVnt,vv

where each nt,v is an integer. Define Wt={vV:nt,v0}. Define

Vt=Wt{t},

and observe that Vt is a finite Gt-invariant subset of V. The edge set Et is defined as the following subset of Vt×,

Et={v×j:vVt and 1j|nt,v|}.

For e=v×j define e+=v and e-=t if nt,v>0, and e+=t and e-=v if nt,v<0. Since each edge of Et has t as one of its endpoints, Γt=(Vt,Et) is connected (when considered as a 1-complex). The Gt-action on Vt induces an action on Et by

g.(v×j)=g.v×j

respecting the adjacency relation. Since ϵp(t)=0 and ϵ is the augmentation map, we have that vVtnt,v=0. Therefore,

eEt(e+-e-)=vVtnt,v(v-t)=p(t).

Step 2.

Let K be a subgroup of G, and let E be a K-set. Then there is G-set E^ and an injective and K-equivariant map ı:EE^ with the following properties:

  1. The map ı induces a bijection of orbit spaces E/KE^/G.

  2. For each eE, the K-stabilizer Ke equals the G-stabilizer Gı(e).

  3. If V is a G-set and f:EV is K-equivariant, then there is a unique G-map f^:E^V such that f^ı=f.

Proof.

Observe that E is isomorphic to the K-space iIK/Ki where I is a complete set of representatives of K-orbits of E and K/Ki denotes the K-space of left cosets of Ki in K. Define E^ as iIG/Ki and observe that the assignment KiKi induces an injective K-map EE^. The three statements of the lemma are observations. ∎

Step 3.

There is a directed G-graph Γ with vertex set V with the following properties:

  1. For each K, the fixed-point set ΓK is a non-empty connected subgraph of Γ.

  2. Γ has finitely many orbits of vertices and edges.

  3. All isotropy groups of Γ belong to .

In particular, as a complex, Γ is an F1,F-space for G.

Proof.

Let Γ be the directed G-graph whose vertex set is the G-set V, and edge G-set E is defined as follows.

Let RT be a complete set of representatives of G-orbits of T. For tRT, let Γt=(Vt,Et) be the directed Gt-graph of Step 1. Let E^t be the G-set provided by Step 2 for the Gt-set Et. The Gt-maps (-)t:EtV and (+)t:EtV defining initial and terminal vertices extend to G-maps (-)t:E^tV and (+)t:E^tV. Let E be the disjoint union of G-sets E=tRTE^t and let (+):EV and (-):EV be the induced maps. Note that E has finitely many G-orbits.

By construction, the group G acts on the directed graph Γ=(V,E) with finitely many orbits of vertices and edges. Moreover, the G-action on Γ as a complex has no inversions (if an edge is fixed setwise, the it is fixed pointwise). The stabilizer Ge of an edge eE is the intersection of two subgroups in , hence Ge. It remains to show that the fixed-point sets ΓK are non-empty and connected for K.

Let h:[T][E] be the morphism of G-modules given by teEte for tRT. Note that h is well defined since the subgroup Gt preserves Et setwise. Since

eEt(e+-e-)=p(t)

by the definition of Et, it follows that for each K the following diagram commutes:

where is the standard boundary map from 1-chains to 0-chains of Γ. Since the top row is an exact sequence, it follows that H0(ΓK,) and hence ΓK is non-empty and connected. ∎

Proof of Corollary 3.5.

Consider the G-set V=P𝒫G/P. Observe that V/G is finite, every isotropy group of V is in , and for every K the fixed-point set VK is non-empty. The first two properties imply that [,V] is a finitely generated free 𝒪(G)-module, and the third property implies that the augmentation map

[,V]ϵG,

is surjective. Since G is FP1,, there is a partial FP1,-resolution

[,T]𝑝[,V]ϵG,0.

Then the proof of Proposition 3.3 provides an F1,-graph Γ with vertex set V. Since 𝒫 is an almost malnormal collection, the intersection of the G-stabilizers of two distinct vertices is finite, and hence stabilizers of edges of Γ are finite. ∎

3.2 Coned-off Cayley complexes and F2,

For this subsection, let G be a group, let 𝒫 be a collection of subgroups of G, and let be the family generated by 𝒫.

Definition 3.6 (Finite relative presentations [27]).

The group G is finitely presented relative to 𝒫 if there is finite subset S of G, and a finite subset R of words over the alphabet SP𝒫P such that the natural homomorphism from the free product F=(P𝒫P)F(S) into G is surjective and its kernel is normally generated by the elements of R (considered as elements of F); here F(S) denotes the free group on the set of letters S. In this case, the data S,𝒫| is called a finite relative presentation of G.

Definition 3.7 (Cayley graph).

Let S be a subset of G closed under inverses. The directed Cayley graphΓ=Γ(G,S) is the directed graph with vertex set G and edge set define as follows: for each gG and sS, there is an edge from g to gs labelled s. It is an observation that if S generates G, then Γ is connected, and that if S is finite then the quotient Γ/G is a finite graph.

The undirected Cayley graph is obtained from Γ(G,S) by first ignoring the orientation of the edges, and then identifying edges between the same pair vertices. Observe that any combinatorial path in the undirected Cayley graph is completely determined by its starting vertex and a word in the alphabet S. If the path is locally embedded, the corresponding word is reduced. From here on, we will only consider the undirected Cayley graph which we will denote by Γ(G,S), and we will call it the Cayley graph of G with respect to S.

The definition of coned-off Cayley complex as well as Lemma 3.9 below are based on [17, Definition 2.47, Lemma 2.48], respectively. In the cited article, the definition of coned-off Cayley complex has some minor differences, and the constructions assume that each P𝒫 is finitely generated. We do not require this last assumption for our arguments

Definition 3.8 (Coned-off Cayley graph and coned-off Cayley complex).

Let S,𝒫| be a finite relative presentation of G.

The coned-off Cayley graph Γ^ of G relative to 𝒫 and S is the G-graph obtained from the Cayley graph of G with respect to SS-1, by adding a new vertex v(gP) for each left coset gP with gG and P𝒫, and edges from v(gP) to each element of gP. The vertices of the form v(gP) are called cone-vertices, and any other vertex a non-cone vertex. The G-action on the cone-vertices is defined using the G-action on the corresponding left cosets, and this action extends naturally to the edge set.

Observe that each locally embedded path in Γ^ between elements of G has a unique label corresponding to a word in the alphabet S𝒫P, since edges between vertices in G have a label in SS-1, and for each oriented and embedded path of length two with middle vertex a cone point v(gP) corresponds a unique element of P. The label of a path that consists of a single non-cone vertex is defined as the empty word.

The coned-off Cayley complex X induced by S,𝒫| is the 2-complex obtained by equivariantly attaching 2-cells to the coned-off Cayley graph Γ^ as follows. Attach a 2-cell with trivial stabilizer to each loop of Γ^ corresponding to a relator r in a manner equivariant under the G-action on Γ^.

Lemma 3.9.

Let S,P|R be a finite relative presentation of G. The induced coned-off Cayley complex X is simply-connected.

Proof.

Let T be the coned-off Cayley graph of F=(P𝒫P)F(S) with respect to 𝒫 and S. Observe that T is simply-connected since it is a tree. Let Γ^ be the coned-off Cayley graph of G with respect to 𝒫 and S. Observe that Γ^ is recovered as the quotient T/N. The main observation is that TΓ^ is a normal covering space with N as a group of deck transformations. Indeed, the edge stabilizers of T with respect to N are trivial by definition; the vertex stabilizers of T with respect to N are trivial since each factor P of F embeds in G an therefore NP is trivial for every P𝒫. It follows that the 1-skeleton of X has fundamental group isomorphic to N, and since N is the normal closure of {r1,,rm} in F the definition of X implies that π1(X) is trivial. ∎

Proposition 3.10.

Suppose that P is a finite and malnormal collection of subgroups of G, and S,P|R is a finite relative presentation of G. Then the induced coned-off Cayley complex X is an F2,F-space with trivial 1-cell G-stabilizers. In particular, for each KF the fixed-point set XK is either a point or X.

Proof.

The space X is simply-connected by Lemma 3.9, and since the relative presentation is finite, the G-action on X is cocompact. Let K be a non-trivial subgroup. Since the G-stabilizers of 1-cells and 2-cells of X are trivial, the fixed-point set XK consists only of 0-cells. Since 𝒫 is a malnormal collection and K fixes at least one cone-vertex, it follows that XK is a single cone-vertex of X. Therefore X is an F2,-space with trivial 1-cell stabilizers. ∎

4 The relative homological Dehn function

Throughout this section, G is a group, is a family of subgroups, and G is FP2,.

Definition 4.1.

Suppose that F is a free G-module with G-basis B. Then the set {gb:gG,bB} is a basis for F as a free -module inducing a G-equivariant 1-norm 1. We call a free G-module based if it is understood to have a fixed G-basis and we use this basis for the induced 1-norm 1.

Definition 4.2.

Let η:FM be a surjective homomorphism of G-modules and suppose that F is free, finitely generated, and based. The filling norm on M induced by η and the based free module F is the G-equivariant norm defined by

mη=min{x1:xF,η(x)=m}.

Definition 4.3 (Algebraic definition of relative homological filling function).

The homological filling function of G relative to is the function

FVG,:

defined as follows. Let P be a resolution of 𝒪(G)-modules for G, of type FP2, and let

P22P11P00

be the resolution of finitely generated G-modules obtained by evaluating P at G/1. Choose filling norms for the G-modules P1 and P2, denoted by P1 and P2, respectively. Then

FVG,(k)=max{γ2:γ𝖪𝖾𝗋(1),γP1k}

where 2 is the filling norm on 𝖪𝖾𝗋(1) given by

γ2=min{μP2:μP2,2(μ)=γ}.

Theorem 4.4.

The growth rate of FVG,F is an invariant of the pair (G,F).

The rest of the section is divided into two parts, first the proof of Theorem 4.4, and then a discussion on a topological approach to FVG,.

4.1 Proof of Theorem 4.4

The proof in the case that consists only of the trivial subgroup was proved in [19, Theorem 3.5]. The argument below is essentially a translation of the argument in [19, Proof of Theorem 3.5] from the category of G-modules to the category of 𝒪(G)-modules up to using Lemma 4.6 instead of [19, Lemma 2.8].

Lemma 4.5 ([14, Lemma 4.1], [19, Lemma 2.9]).

Any two filling norms η and θ on a finitely generated ZG-module M are equivalent in the sense that there exists a constant C0 such that C-1mηmθCmη for all mM.

Lemma 4.6.

Suppose that φ:PQ is a homomorphism between finitely generated ZG-modules. Let P and Q denote filling norms on P and Q, respectively. There exists a constant C>0 such that φ(p)QCpP for all pP.

Proof.

Consider the commutative diagram

constructed as follows. Let A and B be finitely generated and based free G-modules, and let AP and BQ be surjective morphisms inducing the filling norms P and Q. Since A is free and BQ is surjective, there is a lifting φ~:AB of φ. Since φ~ is morphism between finitely generated, free, based G-modules, by [19, Lemma 2.7], there is a constant C such that

φ~(a)1Ca1

for every aA. Let pP and let aA such that ρ(a)=p. It follows that

φ(p)Qφ~(a)1Ca1.

Since the above inequality holds for any aA with ρ(a)=p, it follows that

φ(p)QCminρ(a)=p{a1}=CpP.

Proof of Theorem 4.4.

Let P and Q be two resolutions of 𝒪(G)-modules for G, of type FP2,. For i=1,2, let Pi and Qi be filling norms of the components Pi[G/1] and Qi[G/1], respectively. Denote by FVG,P and FVG,Q the homological filling functions induced by P and Q, respectively. By symmetry, it is enough to prove that there is C0 such that for every k,

(4.1)FVG,Q(k)CFVG,P(Ck+C)+Ck+C.

Define C as follows. Since the category of 𝒪(G)-modules is an abelian category with enough projectives, any two projective resolutions of an 𝒪(G)-module are chain homotopy equivalent. Therefore, there are chain maps fi:QiPi, gi:PiQi, and a map hi:QiQi+1 such that

i+1hi+hi-1i=gifi-Id.

From here on, we only consider the components at G/1 of these resolutions and morphisms, so we are in the category of G-modules. We have two exact sequences of finitely generated G-modules

P2δ2P1δ1P00,Q22Q11Q00

and the morphisms fi:QiPi, gi:PiQi, and hi:QiQi+1. Let C be the maximum of the constants given by Lemma 4.6 for g2, h1, and f1 and the filling-norms provided.

Inequality (4.1) is proved exactly as in [19, Proof of Theorem 3.5] by using Lemma 4.6 instead of [19, Lemma 2.8]. We include the short argument for completeness.

Fix k. Let α𝖪𝖾𝗋(1) be such that αQ1k. Choose βP2 such that δ2(β)=f1(α) and f1(α)δ2=βP2. Then

α=g1f1(α)-2h1(α)+h01(α)=g1δ2(β)-2h1(α)=2g2(β)-2h1(α)=2(g2(β)-h1(α)).

It follows that

α2g2(β)-h1(α)Q2
g2(β)Q2+h1(α)Q2
CβP2+CαQ1
=Cf1(α)δ2+CαQ1
CFVG,P(f1(α)P1)+CαQ1
CFVG,P(CαQ1)+CαQ1
CFVG,P(Ck)+Ck.

Since α was arbitrary, inequality (4.1) holds for every k completing the proof. ∎

4.2 Topological approach to FVG,

Definition 4.7.

Let X be a cell complex. The homological Dehn function of X is the function FVX:{} defined as

FVX(k)=max{γ:γZ1(X,),γ1k}

where

γ=min{μ1:μC2(X,),(μ)=γ}

and the maximum and minimum of the empty set are defined as zero and , respectively.

Recall that a cell complex X is n-acyclic if its reduced homology groups with integer coefficients H~k(X,) are trivial for 0kn.

Definition 4.8.

The group G is FHn, if there is a cocompact G-complex X such that all isotropy groups of X belong to and for every K the fixed-point set XK is a non-empty (n-1)-acyclic subcomplex of X. Such a complex is called an FHn,-complex.

Definition 4.9 (Topological definition of relative homological Dehn function).

Let X be an FH2,-complex for G. The homological filling function FVG, of G relative to , is the (equivalence class of the) homological filling function FVX:{}.

Proposition 4.10.

Suppose that G is of type FH2,F. Then the algebraic and topological definitions of relative homological Dehn function are equivalent.

Proof.

Let X be an FHn,-complex for G, see Definition 4.8. It follows that the induced augmented cellular chain complex of 𝒪(G)-modules C(X,𝒪(G)) is an FP2, resolution of G,. By Theorem 4.4, we have that FVX and FVG, are equivalent. ∎

Proposition 4.11.

Suppose that G is of type FP2,F.

  1. If X is an F1,-graph, then there is a 1 -acyclic F1,-space Y with 1 -skeleton X and finitely many G-orbits of 2 -dimensional cells.

  2. If Y is a 1 -acyclic F1,-space, then FVG, and FVY are equivalent.

The proof of Proposition 4.11 requires the following lemma.

Lemma 4.12.

Suppose that G is of type FP2,F and X is an F1,F-space. Then the OF(G)-module of cellular 1-cycles Z1(X,OF(G)) is finitely generated.

Proof.

Consider the exact sequence of 𝒪(G)-modules

0Z1(X,𝒪(G))C1(X,𝒪(G))C0(X,𝒪(G))G,0.

The assumptions on X imply that each Ci(X,𝒪(G)) is finitely generated and free. Since G is FP2,, there is an exact sequence of 𝒪(G)-modules

0KP1P0G,0

where each Pi is finitely generated and projective, and K is finitely generated. Applying Schanuel Lemma 2.2 to these sequences shows that Z1(X,𝒪(G)) is finitely generated. ∎

Proof of Proposition 4.11.

We claim that there are finitely many circuits, say γ1,,γn, of X such that the corresponding 1-cycles form a generating set of the G-module Z1(X,). By Lemma 4.12 the O(G)-module of cellular 1-cycles Z1(X,𝒪(G)) is finitely generated, hence Z1(X,) is finitely generated. The claim follows by observing that each cellular 1-cycle of X can be expressed as a finite sum of 1-cycles iαi where each αi is induced by a circuit [15, Lemma A2]. Let Y by the 2-dimensional G-complex with 1-skeleton X obtained as follows. For each circuit γi, attach to X a 2-dimensional cell with boundary γi and trivial G-stabilizer, and extend equivariantly. By construction, Y is an F1,-space with trivial H1(Y,), and finitely many G-orbits of 2-cells.

To prove the second statement, Lemma 4.12 implies that Z1(Y,𝒪(G)) is finitely generated. Hence there is an FP2, resolution of the form

Q2C1(Y,𝒪(G))C0(Y,𝒪(G))G,0

where Q2 is finitely generated and free. The expression defining FVG, using this resolution and the definition of FVY differ only on the chosen filling norm on Z1(Y,). Since any two filling norms are equivalent, Lemma 4.5, we have that FVG, and FVY are equivalent. ∎

5 Proof of Theorem 1.14

The section is divided into two parts. In the first part, we introduce the notion of submodule distortion and prove a technical result relating retractions and module distortion, Proposition 5.6. In the second part, this proposition is used to prove Theorem 1.14. Through the rest of the section G is a group and is a family of subgroups.

5.1 Submodule distortion and retractions

Definition 5.1.

The distortion function of a finitely generated G-submodule P in a finitely generated G-module L is defined by

𝖣𝗂𝗌𝗍PL(k)=max{γP:γP and γLk}

where P and L are filling norms for P and L, and 𝖣𝗂𝗌𝗍PL(k) is allowed to be .

Remark 5.2.

The equivalence of filling norms on a finitely generated G-module, Lemma 4.5, implies that the growth rate of 𝖣𝗂𝗌𝗍PL(k) is independent of the choice of filling norms.

Remark 5.3.

In the framework of Definition 4.3, the homological filling function of G relative to is the distortion function of 𝖪𝖾𝗋(1) in P1, specifically, FVG,=𝖣𝗂𝗌𝗍𝖪𝖾𝗋(1)P1.

Definition 5.4.

A G-module M is -free if there is a G-set S with all isotropy groups in such that the induced G-module [S] is isomorphic to M.

Remark 5.5.

Let M be a free 𝒪(G)-module. Then the G/1-component M[G/1] is an -free G-module. If M is finitely generated as an 𝒪(G)-module, then M[G/1] is finitely generated as a G-module.

Proposition 5.6.

Let H be a subgroup of G. Consider a commutative diagram

where

  1. L and P are finitely generated G-modules and P is a submodule of L,

  2. M and Q are finitely generated H-modules and Q is a submodule of M,

  3. as H-modules, Q is a submodule of P, and M is a submodule of L,

  4. all the arrows in the diagram are inclusions.

Let L and M be filling norms on L and M, respectively. Suppose there is an integer C01 such that for every xM,

xLC0xM.

Suppose that P is F-free and there is a commutative diagram of ZH-modules

Then

𝖣𝗂𝗌𝗍QM𝖣𝗂𝗌𝗍PL.

Proof of Proposition 5.6.

Since P is a finitely generated -free G-module, there is a G-set S with isotropy groups in and finitely many G-orbits such that P[S]. Denote by P the 1-norm on P induced by the free -basis S; observe that this norm is a filling norm on P as a G-module.

Since Q is a finitely generated H-module, there is a partition S=S1S2 where each Si is an H-equivariant subset of S, the quotient S1/H is finite, and the inclusion ı:QP factors through the inclusion P1P as in the commutative diagram

where P1=[S1], and P2=[S2]. Equip the H-module P1 with the 1-norm P1 induced by the free -basis S1, and observe that this norm is a filling norm on P1 as a finitely generated H-module. We have that

xP1=xP

for every xP1. Moreover, if p=xy is an element of PP1P2, we have that

xPpP.

Observe that we can assume that ρ:PQ restricts to the trivial morphism when restricted to P2. Simply redefine ρ on P2 as the zero morphism and observe that the commutative diagram of the statement of the proposition still holds.

Consider the morphism of finitely generated H-modules ρ:P1Q. By Lemma 4.6 there is an integer C1 such that for every xP1,

ρ(x)QC1xP1.

Let pP be arbitrary and suppose p=xy where xP1 and yP2. Then

ρ(p)Q=ρ(x)QC1xP1=C1xPC1pP

where the first equality follows from ρ being trivial on P2.

Let C=max{C0,C1}. Let xQ be an arbitrary element and observe that

xQ=ρı(x)QCı(x)PC𝖣𝗂𝗌𝗍PL(ı(x)L)C𝖣𝗂𝗌𝗍PL(Cı(x)M)

where the last inequality follows from the hypothesis that xLCxM for every xM. Therefore for every positive integer k,

𝖣𝗂𝗌𝗍QM(k)C𝖣𝗂𝗌𝗍PL(Ck).

5.2 Proof of Theorem 1.14

Definition 5.7.

The Bredon cohomological dimension of G relative to is at most n, denoted by 𝖼𝖽(G)n, if there is a projective resolution of 𝒪(G)-modules

0PnP1P0𝔾,0.

Proposition 5.8.

If G is of type FF and cdF(G)2, then there is an F1,F-complex X such that Z1(X,OF(G)) is finitely generated and free.

The proof of Proposition 5.8 requires the following definition and lemma.

Definition 5.9.

An 𝒪(G)-module M is stably free if there is a finitely generated free 𝒪(G)-module F such that MF is a free 𝒪(G)-module.

Lemma 5.10.

Let X be a G-graph and let v be a vertex of X with G-stabilizer KF. Let X be the G-graph obtained from X by adding a new G-orbit of edges with representative an edge e such that Ge=K and both endpoints are equal to v. Then the OF(G)-module of cellular 1-cycles Z1(X) is isomorphic to Z1(X)Z[,G/K].

Proof.

For each subgroup J, the component Z1(X)[G/J] is identified with the -module of cellular 1-cycles of the fixed-point set XJ. Hence there are natural isomorphisms Z1(X)[G/J]Z1(XJ) if J is not a subgroup of a conjugate of K, and Z1(X)[G/J]Z1(XJ) if J is a subgroup of a conjugate of K. One verifies that these isomorphisms of -modules are the components of an isomorphism of 𝒪(G)-modules Z1(X)Z1(X)[,G/K]. ∎

Proof of Proposition 5.8.

Let X be a cocompact model for E(G), and let n be its dimension. Consider the exact sequences of 𝒪(G)-modules

(5.1)0Z1(X)C1(X)C0(X)G,0

and

(5.2)0Cn(X)C1(X)C0(X)G,0.

The assumptions on the model X imply that each Ci(X) is a finitely generated and free 𝒪(G)-module. Since 𝖼𝖽(G)2, there is an exact sequence of projective 𝒪(G)-modules

(5.3)0P2P1P0G,0.

Applying Schanuel’s lemma to the exact sequences (5.1) and (5.3) implies that Z1(X) is projective. Then Schanuel’s lemma can be applied to the exact sequences (5.1) and (5.2), which yields that Z1(X) is stably free and finitely generated. Let Y be the complex obtained by adding finitely many new orbits of edges to X in such a way that Lemma 5.10 implies that Z1(Y) is a finitely generated free 𝒪(G)-module. Then Y is an F1,-space for G. ∎

Proposition 5.11.

Let Γ be an F1,F-complex for G. If H is a subgroup of G of type F1,FH, then Γ contains an H-equivariant subgraph Δ which is an F1,FH-complex.

Proof.

Since the subgroup H is FP1,H, Proposition 3.3 implies that there is an F1,H-complex Δ. Let {vi:iI} and {ej:iJ} be complete sets of representatives of H-orbits of vertices and edges of Δ, respectively. For each iI, choose a vertex wi in the fixed-point set ΓHvi which is non-empty since Hvi. The set assignment viwi induces an H-map ϕ from the vertex set of Δ to the one of Γ. For each jJ, let Hj be the (pointwise) H-stabilizer of ej and let v- and v+ be the endpoints of ej. Let γj be an edge path in the fixed-point set ΓHj from ϕ(v-) to ϕ(v+); ΓHj is connected since Hj. After subdividing Δ, the assignment ejγj extends ϕ to an H-map of graphs ϕ:ΔΓ. Replacing Δ with the image of ϕ proves the claim. ∎

Proof of Theorem 1.14.

By Proposition 5.8, there is an F1,-complex Γ for G such that Z1(Γ,𝒪(G)) is free and finitely generated. By Proposition 5.11, Γ contains an H-equivariant subgraph Δ which is an F1,H-complex for H. Since H is FP2,H, we have that Z1(Δ,𝒪H(H)) is finitely generated.

From here on, all considered modules are 𝒪H(H)-modules. Consider Γ and Δ as H-graphs. Then there is a short exact sequence of cellular 𝒪H(H)-chain complexes

(5.4)0C(Δ)C(Γ)C(Γ,Δ)0

where Ci(Γ,Δ) is the quotient Ci(Γ)/Ci(Δ). Consider the induced long exact homology sequence. Observing that H2(Γ,Δ) is trivial, and that the reduced homology H~0(Δ) is trivial (since ΔK is connected for all KH), we obtain that the sequence

(5.5)0Z1(Δ)Z1(Γ)Z1(Γ,Δ)0

is exact. Considering once more the long exact sequence induced by (5.4), one verifies that

(5.6)0Z1(Γ,Δ)C1(Γ,Δ)C0(Γ,Δ)0

is an exact sequence of 𝒪H(H)-modules. Since Δ is an H-equivariant subgraph of Γ, we have that Ci(Δ) is a free factor of Ci(Γ); hence Ci(Γ,Δ) is free. Therefore, the sequence (5.6) implies that Z1(Γ,Δ) is projective. The exact sequence (5.5) together with Z1(Γ,Δ) being projective implies that there is a commutative diagram of 𝒪H(H)-modules

The proof of the proposition concludes by invoking Proposition 5.6 as follows. Considering the H/1-component of the above commutative diagram, we obtain the analogous diagram for the H-modules of integral cellular 1-cycles of Γ and Δ. Since Z1(Γ,𝒪(G)) is finitely generated and free, it follows that P:=Z1(Γ,) is a finitely generated and -free G-module. Analogously, since Z1(Δ,𝒪H(H)) is finitely generated, Q:=Z1(Δ,) is a finitely generated H-module. Moreover, Q is a submodule of the finitely generated H-module M:=C1(Δ,), and analogously P is a submodule of the finitely generated G-module L:=C1(Γ,). Observe that we can assume that ML. Considering the natural 1-norms L and M on L and M induced by the collection 1-cells (with a chosen orientation) of Γ and Δ, respectively, we have the equality σL=σM for every σM. Then Remark 5.3 implies

FVH,H=𝖣𝗂𝗌𝗍QMandFVG,=𝖣𝗂𝗌𝗍PL.

By Proposition 5.6, FVH,HFVG,. ∎

6 Relatively hyperbolic groups

6.1 Proof of Theorem 1.12 (1)

Let G be a group and let 𝒫 be a collection of subgroups generating a family . Suppose that 𝒫 is a finite and almost malnormal collection, that G is FP2,, and that FVG,(n) has linear growth.

By Corollary 3.5, there is an F1,-graph X with finite edge stabilizers. By Proposition 4.11, there is an F1,-space Y with 1-skeleton X, with finitely many G-orbits of 2-dimensional cells, and such that FVG, and FVY are equivalent. To conclude that G is hyperbolic relative to , it is enough to prove that X is fine and hyperbolic. Since FVG, has linear growth, the proof concludes by invoking the following result.

Lemma 6.1 ([23, Corollary 3.3]).

Let Y be a 1-acyclic 2-dimensional cell complex such that there is a bound on the length of attaching maps of 2-cells. If FVY(n) has linear growth, then the 1-skeleton of Y is a fine hyperbolic graph.∎

6.2 A class of relatively hyperbolic groups of Bredon cohomological dimension 2

This last subsection relies on small cancellation theory over free products; our main reference is Lyndon–Schupp textbook on Combinatorial Group Theory [22, Chapter V, Section 9]. The following remark recalls some well known facts on small cancellation products.

Remark 6.2.

Suppose that 𝒫={Pi} is a finite collection of groups, let F=Pi be the corresponding free product, and let R be a symmetrized subset of F satisfying the C(1/6)-small cancellation condition. Let N be the normal closure of R in F.

  1. The collection 𝒫 can be regarded as a collection of subgroups of F/N. Indeed, Greendlinger’s lemma [22, Chapter V, Theorem 9.3] implies that N does not contain elements of length less than three. In particular, the natural map FF/N embeds each factor Pi of F, and no two factors are identified.

  2. The group F/N is hyperbolic relative to 𝒫; this is a direct consequence of [22, Chapter V, Theorem 9.3] using Osin’s approach to relative hyperbolicity [27, p. 4, (II)].

  3. If each P𝒫 is torsion-free, then 𝒫 is malnormal in F/N, see [27, Proposition 2.36].

Theorem 6.3.

Suppose that P={Pi} is a finite family of torsion-free groups, that R is a finite symmetrized subset of the free product F=Pi satisfying the C(1/6)-small cancellation condition and with no proper powers, and that G=F/N where N is the normal closure of R.

If X is the coned-off Cayley complex of a relative presentation P|R where R is a minimal subset of R with normal closure N in F, then X is a cocompact EFG-complex where F is the family generated by P. In particular, gdF(G)2.

Proof.

In view of Proposition 3.10 and Remark 6.2, it is left to prove that X is contractible. We argue that every spherical diagram SX is reducible as explained below. It follows that X has trivial second homotopy group by a remark of Gersten [13, Remark 3.2]. Since X is 2-dimensional and simply-connected, an application of Hurewicz and Whitehead theorems from algebraic topology imply that X is contractible.

A spherical diagram is a combinatorial cellular map SX where S is topologically the 2-sphere. Roughly speaking, the diagram is reducible in the sense of Gersten if there are two distinct faces (2-cells) of the 2-sphere S that have an edge in common and map to the same 2-cell of X by a mirror image, see [13, Section 3.1] for a precise definition. The small cancellation condition implies that every spherical diagram SX is reducible; we sketch an argument below following Ol’shanskiĭ [26, Proof of Theorem 13.3].

Let γ be a path in X whose endpoints are non-cone vertices. We observed in Definition 3.8 that γ has as a label a reduced word in the alphabet Pi. If α is a subpath of γ, then the label of α is defined as the label of the minimal length subpath of γ such that its endpoints are non-cone vertices and it contains α as a subpath. In particular, if α is a trivial path then either α is a non-cone vertex and its label is the empty word, or α is a cone-vertex v(gP) and its label is a non-trivial element of the group P𝒫.

Now we remark two direct consequences of Greendlinger’s lemma [22, Chapter V, Theorem 9.3] together with their interpretations/consequences on the cell structure of X. We leave some of the details to the reader.

  1. If r has semi-reduced form r=ab and a is not the empty word, then bN.

    As a consequence, any closed curve γX in the 1-skeleton of X with label r is an embedded circle.

  2. Let r,r. If r and r have semi-reduced forms asbt and aubv, respectively, and su-1 is a reduced form such that su-1N, then r=r.

    As a consequence, if D and D are a pair of 2-cells of X whose boundaries intersect, then either D=D or their boundaries intersect in a single path (possibly a trivial path) whose induced label (as a subpath of the boundary of D) is a piece.

    This last statement relies on the minimality of and the assumption that has no proper powers, and not only on the stated consequence of Greendlinger’s lemma. Suppose that there is a pair of distinct 2-cells D and D of X such that their boundary intersection contains a path whose label is not a piece. Then from the definition of piece, it follows that the labels r and r of the boundaries of D and D, respectively, are the same reduced word up to taking the inverse and a cyclic conjugation. By the minimality of , the words r and r correspond to the same element of , say r. By the definition of X, since D and D are distinct and have the same boundary path labelled by r, the word r is a proper power. But this contradicts that has no proper powers.

    In particular, the consequence of Greendlinger’s lemma is only used to show that if the boundary intersection of D and D is non-empty, then it is connected. This is obtained by observing that for a suitable orientations of the boundary paths of D and D if their intersection is not connected their labels have semi-reduced forms asbt and aubv, respectively, and such that su-1 is the label of an embedded closed path.

Suppose that SX is a spherical diagram that is not reducible. Let Φ be the graph whose vertices are the 2-cells of S; and given two vertices, there is an edge between them for each connected component of the intersection of the boundary paths of the corresponding 2-cells. It is immediate that Φ is planar. Below we argue that every vertex of Φ has degree at least six, and that Φ is simplicial, i.e., no double edges and no loops. Since every finite planar simplicial graph has a vertex of degree at most five, the existence of Φ yields a contradiction. Hence a non-reducible spherical diagram SX cannot exist.

The first bullet statement above implies that Φ has no edges that connect a vertex to itself. Since SX is not reducible, the second bullet statement above implies that if D and D are two distinct 2-cells of S and their boundaries have non-empty intersection, then they intersect in a single path whose induced label is a piece. This implies that Φ has no two distinct edges connecting the same vertices. Hence Φ is simplicial. Finally, the C(1/6)-condition implies that every r cannot be the concatenation of less than six pieces, and we observed that the intersection of any pair of 2-cells of S is along a path with label a piece; it follows that every vertex of Φ has degree at least six. ∎


Communicated by Pierre-Emmanuel Caprace


Funding statement: The author acknowledges funding by the Natural Sciences and Engineering Research Council of Canada, NSERC.

Acknowledgements

The author thanks Gaelan Hanlon, Tomaz Prytuła and Mario Velasquez for comments. We would also like to thank Dieter Degrijse for pointing out an error in an earlier version of the article, Ian Leary for consultation at the Conference on Finiteness Conditions in Topology and Algebra 2015 and Denis Osin for a conversation suggesting the example in Remark 1.7 at the Geometric and Asymptotic Group Theory with Applications Conference 2016. We specially thank the referees for suggestions, feedback, and pointing out necessary corrections.

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Received: 2016-5-6
Revised: 2017-5-30
Published Online: 2017-7-6
Published in Print: 2017-11-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

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