François Dahmani

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.

2000 Mathematical Subject Classification: 20F67, 20F69.

Keywords: relatively hyperbolic group, classifying space, Rips complex, Z-structure.

E-mail: dahmani@math.u-strasbg.fr