Boundaries of geometrically finite groups

Abstract

We show that the limit set of a relatively hyperbolic group with no separating horoball is locally connected if it is connected. On the other hand, if there is a separating horoball centred on a parabolic point, one obtains a non-trivial splitting of the group over a parabolic subgroup relative to the maximal parabolic subgroups. Together with results from elsewhere, one deduces that if $ \Gamma $ is a relatively hyperbolic group such that each maximal parabolic subgroup is one-or-two ended, finitely presented, and contains no infinite torsion subgroup, then the boundary of $ \Gamma $ is locally connected if it is connected. As a corollary, we see that the limit set of a geometrically finite group acting on a complete simply connected manifold of pinched negative curvature must be locally connected if it is connected.