A Model for the Universal Space for Proper Actions of a Hyperbolic Group

TL;DR

This paper proves that the Rips complex associated with a hyperbolic group serves as a finite universal space for proper actions, and shows hyperbolic groups have finitely many conjugacy classes of finite subgroups.

Contribution

It establishes the contractibility of fixed point sets in the Rips complex and confirms it as a finite model for the universal space of proper actions in hyperbolic groups.

Findings

Fixed point sets are contractible for all finite subgroups.

The Rips complex is a finite model for the universal space of proper actions.

Hyperbolic groups have finitely many conjugacy classes of finite subgroups.

Abstract

Let $G$ be a word hyperbolic group in the sense of Gromov and $P$ its associated Rips complex. We prove that the fixed point set $P^H$ is contractible for every finite subgroups $H$ of $G$. This is the main ingredient for proving that $P$ is a finite model for the universal space $e.g.$ of proper actions. As a corollary we get that a hyperbolic group has only finitely many conjugacy classes of finite subgroups.