Cohomology and the Bowditch Boundary

Abstract: We give a group cohomological description of the \v{C}ech cohomology of the Bowditch boundary of a relatively hyperbolic group pair, generalizing a result of Bestvina-Mess about hyperbolic groups. In case of a relatively hyperbolic Poincar\'e duality group pair, we show the Bowditch boundary is a homology manifold. For a three-dimensional Poincar\'e duality pair, we recover the theorem of Tshishiku-Walsh stating that the boundary is homeomorphic to a two-sphere.