Codimension-1 Subgroups and Splittings of Groups Original Research Article

We show that under certain circumstances, a codimension-1 subgroupHof a finitely generated groupGeither provides a splitting ofGas an amalgam or provides a codimension-1 subgroup ofH. In particular, ifGis hyperbolic andHis 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.