Topology and Geometric Group Theory Seminar
Hyperbolic groups are often studied by their Cayley graphs which are not well-defined up to homotopy. These Cayley graphs can be compactified by attaching a boundary which turns out to be well-defined up to homeomorphism. Thus, one might hope for a relation between a hyperbolic group and homotopy invariant properties of its boundary. Indeed, Bestvina and Mess proved an isomorphism between group cohomology and Cech cohomology of the boundary. I will discuss the analogue of this theorem for relatively hyperbolic groups and some of its consequences.