Rigidity and topological conjugates of topologically tame Kleinian groups

Minsky proved that two Kleinian groups and are quasi-conformally conjugate if they are freely indecomposable, the injectivity radii at all points of , are bounded below by a positive constant, and there is a homeomorphism from a topological core of to that of such that and map ending laminations to ending laminations. We generalize this theorem to the case when and are topologically tame but may be freely decomposable under the same assumption on the injectivity radii. As an application, we prove that if a Kleinian group is topologically conjugate to another Kleinian group which is topologically tame and not a free group, and both Kleinian groups satisfy the assumption on the injectivity radii as above, then they are quasi-conformally conjugate.