Codimension-1 Subgroups and Splittings of Groups

Abstract We show that under certain circumstances, a codimension-1 subgroup H of a finitely generated group G either provides a splitting of G as an amalgam or provides a codimension-1 subgroup of H . In particular, if G is hyperbolic and H is quasiconvex, one gets a descending sequence of codimension-1 subgroups, terminating at a splitting. In the process, we settle a conjecture of Kropholler and Roller for the case of quasiconvex subgroups of hyperbolic groups.