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Strong convergence of Kleinian groups and Carathéodory convergence of domains of discontinuity

Published online by Cambridge University Press:  24 October 2008

Ken'ichi Ohshika
Ken'ichi Ohshika
Affiliation:
IHES, 91440 Bures-sur-Yvette, France, and Department of Mathematics, TokyoInstitute of Technology, Oh-okayama, Tokyo 152, Japan
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Extract

In the theory of Kleinian groups, important examples of Kleinian groups are frequently constructed as algebraic limits of known sequences of Kleinian groups, for example quasi-conformal deformations of a Kleinian group. It is an important problem to determine the properties of the limit Kleinian group from information on the sequence converging to the limit. The topological properties of the domain of discontinuity and the limit set of a Kleinian group provide valuable pieces of information about the Kleinian group. It is reasonable to expect that in a good situation, the domain of discontinuity of the limit Kleinian group is the Carathéodory limit of the domains of discontinuity of the sequence, or, equivalently, the limit set of the limit Kleinian group is the Hausdorif limit of the limit sets of the sequence. Our main theorem (Theorem 5) shows that this is true when the sequence strongly converges to the limit, and the Kleinian groups of the sequence and the limit preserve the parabolicity in both directions and satisfy the condition (*) introduced by Bonahon.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 112 , Issue 2 , September 1992 , pp. 297 - 307
Copyright
Copyright © Cambridge Philosophical Society 1992

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