Cohomology and the Bowditch Boundary

Abstract

We give a group cohomological description of the Čech cohomology of the Bowditch boundary of a relatively hyperbolic group pair, generalizing a result of Bestvina–Mess about hyperbolic groups. In the case of a relatively hyperbolic Poincaré duality group pair, we show that the Bowditch boundary is a homology manifold. For a three-dimensional Poincaré duality pair, we recover the theorem of Tshishiku–Walsh stating that the boundary is homeomorphic to a two-sphere.