Rigidity and topological conjugates of topologically tame Kleinian groups

Ohshika, Ken’ichi

doi:10.1090/S0002-9947-98-02073-X

Rigidity and topological conjugates of topologically tame Kleinian groups
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Trans. Amer. Math. Soc. 350 (1998), 3989-4022 Request permission

Abstract:

Minsky proved that two Kleinian groups $G_1$ and $G_2$ are quasi-conformally conjugate if they are freely indecomposable, the injectivity radii at all points of $\mathbf {H}^3/G_1$, $\mathbf {H}^3/G_2$ are bounded below by a positive constant, and there is a homeomorphism $h$ from a topological core of $\mathbf {H}^3/G_1$ to that of $\mathbf {H}^3/G_2$ 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 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.

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