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

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.