RIGIDITY AND TOPOLOGICAL CONJUGATES OF TOPOLOGICALLY TAME KLEINIAN-GROUPS

Minsky proved that two Kleinian groups Gr and Ga are quasi-conformally conjugate if they are freely indecomposable, the injectivity radii at all points of H-3/G(1), H-3/G(2) are bounded below by a positive cons tant, and there is a homeomorphism h from a topological core of H-3/G( 1) to that of H-3/G(2) such that h and h(-1) map ending laminations to ending laminations. We generalize this theorem to the case when G(1) and G(2) are topologically tame but may be freely decomposable under t he same assumption on the injectivity radii. As an application, we pro ve that if a Kleinian group is topologically conjugate to another Klei nian group which is topologically tame and not a free group, and both Kleinian groups satisfy the assumption on the injectivity radii as abo ve, then they are quasi-conformally conjugate.