Subgroups of Relatively Hyperbolic Groups of Bredon Cohomological Dimension 2

TL;DR

This paper extends Gersten's result by proving that relatively hyperbolic groups with Bredon cohomological dimension 2 are closed under finitely presented subgroups, using algebraic methods and homological inequalities.

Contribution

It establishes an analogous closure property for relatively hyperbolic groups of Bredon cohomological dimension 2, broadening understanding of subgroup structures.

Findings

Closure of relatively hyperbolic groups of Bredon cohomological dimension 2 under finitely presented subgroups.

Application to $C'(1/6)$ small cancellation products.

Use of algebraic approach to relative homological Dehn functions.

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 $C'(1/6)$ 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.