Height in splittings of hyperbolic groups

Abstract

Suppose H is a hyperbolic subgroup of a hyperbolic group G. Assume there existsn > 0 such that the intersection of n essentially distinct conjugates of H is always finite. Further assume G splits over H with hyperbolic vertex and edge groups and the two inclusions of H are quasi-isometric embeddings. Then H is quasiconvex in G. This answers a question of Swarup and provides a partial converse to the main theorem of [23].