Cohomology and the Bowditch Boundary

TL;DR

This paper explores the cohomological properties of the Bowditch boundary of relatively hyperbolic groups, generalizing known results and establishing conditions under which the boundary is a homology manifold or a sphere.

Contribution

It provides a cohomological framework for understanding the Bowditch boundary, extending results from hyperbolic groups to relatively hyperbolic groups and Poincaré duality pairs.

Findings

Bowditch boundary's 7ech cohomology described cohomologically

Boundary is a homology manifold for Poincare9 duality group pairs

Boundary is homeomorphic to a 2-sphere in 3-dimensional Poincare9 duality pairs

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.