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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Rigidity and topological conjugates of topologically tame Kleinian groups
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by Ken’ichi Ohshika PDF
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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Additional Information
  • Ken’ichi Ohshika
  • Affiliation: Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Meguro-ku, Tokyo 153, Japan
  • MR Author ID: 215829
  • Email: ohshika@ms.u-tokyo.ac.jp
  • Received by editor(s): July 22, 1994
  • Received by editor(s) in revised form: October 14, 1996
  • © Copyright 1998 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 350 (1998), 3989-4022
  • MSC (1991): Primary 57M50; Secondary 30F40
  • DOI: https://doi.org/10.1090/S0002-9947-98-02073-X
  • MathSciNet review: 1451613