Suppose that a finitely generated group G is hyperbolic relative to a collection of subgroups ℙ = {P1,…,Pm}. Let H1,H2 be subgroups of G such that H1 is relatively quasiconvex with respect to ℙ and H2 is not parabolic. Suppose that H2 is elementwise conjugate into H1. Then there exists a finite index subgroup of H2 which is conjugate into H1. The minimal length of the conjugator can be estimated. In the case, where G is a limit group, it is sufficient to assume only that H1 is a finitely generated and H2 is an arbitrary subgroup of G.
['Oleg Bogopolski', 'Kai-Uwe Bux']
From local to global conjugacy of subgroups of relatively hyperbolic groups