Subgroups of relatively hyperbolic groups of Bredon cohomological dimension 2

Abstract

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 FP 2 ) 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 0 .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.

Homological Dehn functions of groups of type $FP_2$

“…Propositions 2.24 and 2.25 below can also be obtained from Theorem 4.4 and Proposition 4.10 of [Mar17], where the relative homological Dehn functions are studied. We include them here for completeness and ease of the reader.…”

“…For groups of type F P 2 some aspects of homological Dehn functions were studied in [Ger92,HM16,Mar17,FM18]. In this article, we unify all these results into a single framework by considering homological finite presentations (Definition 2.5).…”

Dismantlable Classifying Space for the Family of Parabolic Subgroups of a Relatively Hyperbolic group

“…The function FV G (k) can also be defined from algebraic considerations under the weaker assumption that G is F P 2 , see [12,Section 3]. Analogously, for a group G and a family of subgroups F with a cocompact model for E F G, there is relative homological Dehn function FV G,F (k) whose growth rate is an invariant of the pair (G, F), see [18,Theorem 4.5].…”

“…An analogous characterisation for relatively hyperbolic groups is proved in [18, Theorem 1.11] relying on the following corollary. We remark that a converse of Corollary 1.5 requires an additional condition that {P λ } is an almost malnormal collection, see [18,Theorem 1.11(1) and Remark 1.13].…”

A Note on Fine Graphs and Homological Isoperimetric Inequalities

“…The type of homological functions of Definition 1.2 have been considered in the contexts of relatively hyperbolic groups for example in [9,11,15]. Combining Theorems 1.6, 1.7 and 1.8 allows us to provide a characterization of relatively hyperbolic groups in terms of homological Dehn functions stated as Theorem 1.10 below.…”