Dismantlable classifying space for the family of parabolic subgroups of a relatively hyperbolic group

TL;DR

This paper constructs a cocompact classifying space for a relatively hyperbolic group with torsion, using dismantlability, extending known results from torsion-free cases and providing new methods.

Contribution

It introduces a dismantlability-based approach to build classifying spaces for groups with torsion, generalizing previous torsion-free results.

Findings

Existence of cocompact model for $E_{amily} G$ in the presence of torsion

Application of dismantlability to relatively hyperbolic groups

Extension of known results from torsion-free to torsion cases

Abstract

Let $G$ be a group hyperbolic relative to a finite collection of subgroups $\mathcal P$. Let $\mathcal F$ be the family of subgroups consisting of all the conjugates of subgroups in $\mathcal P$, all their subgroups, and all finite subgroups. Then there is a cocompact model for $E_{\mathcal F} G$. This result was known in the torsion-free case. In the presence of torsion, a new approach was necessary. Our method is to exploit the notion of dismantlability. A number of sample applications are discussed.