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Conformal Geometry and Dynamics

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An explicit counterexample to the equivariant $K=2$ conjecture

Authors: Yohei Komori and Charles A. Matthews
Journal: Conform. Geom. Dyn. 10 (2006), 184-196
MSC (2000): Primary 30F40, 30F60, 32G15, 57M50
Published electronically: August 24, 2006
MathSciNet review: 2261047
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Abstract: We construct an explicit example of a geometrically finite Kleinian group $G$ with invariant component $\Omega$ in the Riemann sphere $\textbf {\hat {C}}$ such that any quasiconformal map from $\Omega$ to the boundary of the convex hull of $\textbf {\hat {C}} - \Omega$ in $\textbf {H^3}$ which extends to the identity map on their common boundary in $\textbf {\hat {C}}$, and which is equivariant under the group of Möbius transformations preserving $\Omega$, must have maximal dilatation $K > 2.002$.

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Additional Information

Yohei Komori
Affiliation: Department of Mathematics, Osaka City University, Osaka 558-8585, Japan

Charles A. Matthews
Affiliation: Department of Mathematics, Southeastern Oklahoma State University, Durant, Oklahoma 74701

Received by editor(s): April 20, 2006
Published electronically: August 24, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.