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Uniform perfectness of the limit sets of Kleinian groups

Author: Toshiyuki Sugawa
Journal: Trans. Amer. Math. Soc. 353 (2001), 3603-3615
MSC (2000): Primary 30F40; Secondary 30F45
Published electronically: May 4, 2001
MathSciNet review: 1837250
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In this note, we show, in a quantitative fashion, that the limit set of a non-elementary Kleinian group is uniformly perfect if the quotient orbifold is of Lehner type, i.e., if the space of integrable holomorphic quadratic differentials on it is continuously contained in the space of (hyperbolically) bounded ones. This result covers the known case when the group is analytically finite. As applications, we present estimates of the Hausdorff dimension of the limit set and the translation lengths in the region of discontinuity for such a Kleinian group. Several examples will also be given.

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Additional Information

Toshiyuki Sugawa
Affiliation: Department of Mathematics, Kyoto University, 606-8502 Kyoto, Japan
Address at time of publication: Department of Mathematics, University of Helsinki, P. O. Box 4 (Yliopistonkatu 5), FIN-00014, Helsinki, Finland

Keywords: Uniformly perfect, Kleinian group, translation length, Hausdorff dimension
Received by editor(s): June 16, 1998
Received by editor(s) in revised form: November 27, 2000
Published electronically: May 4, 2001
Dedicated: Dedicated to Professor Hiroki Sato on the occasion of his sixtieth birthday.
Article copyright: © Copyright 2001 American Mathematical Society