Suppose G is a group that splits as \(A*_C B\) or \(A*_C\) , where A, B, C are relatively hyperbolic groups, the monomorphisms \(C\rightarrow A\) and \(C\rightarrow B\) are quasiisometric embeddings of pairs, and G is hyperbolic relative to a natural collection of subgroups. We show that C has finite relative height in G if and only if C is relatively quasiconvex in G. This extends the work of Pal, giving an exact analogue of a theorem of Mj in the context of relatively hyperbolic groups and partially answers a question of Hruska–Wise.
