Group Splittings and Asymptotic Topology

It is a consequence of the theorem of Stallings on groups with many ends that splittings over finite groups are preserved by quasi-isometries. In this paper we use asymptotic topology to show that group splittings are preserved by quasi-isometries in many cases. Roughly speaking we show that splittings are preserved under quasi-isometries when the vertex groups are fundamental groups of aspherical manifolds (or more generally ‘coarse PD(n)-groups’) and the edge groups are ‘smaller’ than the vertex groups.