Classifying Spaces and Boundaries for Relatively Hyperbolic Groups

Abstract

We prove the following: if a group $\Gamma$ is torsion-free, and relatively hyperbolic (with the Bounded Coset Penetration property), relative to a subgroup admitting a finite classifying space, then $\Gamma$ admits a finite classifying space. In this case, if the subgroup admits a boundary in the sense of $\mathcal{Z}$-structures, we prove that $\Gamma$ admits a boundary. This extends classical results of Rips, and of Bestvina and Mess to the relative case.