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Topologically conjugate Kleinian groups

Author: Ken'ichi Ohshika
Journal: Proc. Amer. Math. Soc. 124 (1996), 739-743
MSC (1991): Primary 30F40, 57M50
MathSciNet review: 1346983
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Abstract: Two Kleinian groups $\Gamma_1$ and $\Gamma_2$ are said to be topologically conjugate when there is a homeomorphism $f:S^2 \rightarrow S^2$ such that $f \Gamma_1 f^{-1}= \Gamma_2$. It is conjectured that if two Kleinian groups $\Gamma_1$ and $\Gamma_2$ are topologically conjugate, one is a quasi-conformal deformation of the other. In this paper generalizing Minsky's result, we shall prove that this conjecture is true when $\Gamma_1$ is finitely generated and freely indecomposable, and the injectivity radii of all points of $\mathbf{H}^3/\Gamma_1$ and $\mathbf{H}^3/\Gamma_2$ are bounded below by a positive constant.

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Ken'ichi Ohshika
Affiliation: Department of Mathematics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152, Japan

Keywords: Kleinian group, topological conjugacy
Received by editor(s): November 17, 1993
Communicated by: Ron Stern
Article copyright: © Copyright 1996 American Mathematical Society