Classifying Spaces and Boundaries for Relatively Hyperbolic Groups

We prove the following: if a group Γ is torsion-free, and relatively hyperbolic (with the Bounded Coset Penetration property), relative to a subgroup admitting a finite classifying space, then Γ admits a finite classifying space. In this case, if the subgroup admits a boundary in the sense of Z-structures, we prove that Γ admits a boundary. This extends classical results of Rips, and of Bestvina and Mess to the relative case. 2000 Mathematics Subject Classification 20F67, 20F69.