Peripheral splittings of groups

We define the notion of a "peripheral splitting" of a group. This is essent ially a representation of the group as the fundamental group of a bipartite graph of groups, where all the vertex groups of one colour are held fixed- the "peripheral subgroups". We develop the theory of such splittings and pr ove an accessibility result. The theory mainly applies to relatively hyperb olic groups with connected boundary, where the peripheral subgroups are pre cisely the maximal parabolic subgroups. We show that if such a group admits a non-trivial peripheral splitting, then its boundary has a global cut poi nt. Moreover, the non-peripheral vertex groups of such a splitting are them selves relatively hyperbolic. These results, together with results from els ewhere, show that under modest constraints on the peripheral subgroups, the boundary of a relatively hyperbolic group is locally connected if it is co nnected. In retrospect, one further deduces that the set of global cut poin ts in such a boundary has a simplicial treelike structure.