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Measurable rigidity for Kleinian groups

Published online by Cambridge University Press:  01 June 2015

WOOJIN JEON  and
KEN’ICHI OHSHIKA
WOOJIN JEON
Affiliation:
School of Mathematics, KIAS, Hoegiro 87, Dongdaemun-gu, Seoul, 130-722, Korea email jwoojin@kias.re.kr
KEN’ICHI OHSHIKA
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan email ohshika@math.sci.osaka-u.ac.jp
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Abstract

Let $G,H$ be two Kleinian groups with homeomorphic quotients $\mathbb{H}^{3}/G$ and $\mathbb{H}^{3}/H$ . We assume that $G$ is of divergence type, and consider the Patterson–Sullivan measures of $G$ and $H$ . The measurable rigidity theorem by Sullivan and Tukia says that a measurable and essentially directly measurable equivariant boundary map $\widehat{k}$ from the limit set $\unicode[STIX]{x1D6EC}_{G}$ of $G$ to that of $H$ is either the restriction of a Möbius transformation or totally singular. In this paper, we shall show that such $\widehat{k}$ always exists. In fact, we shall construct $\widehat{k}$ concretely from the Cannon–Thurston maps of $G$ and $H$ .

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 8 , December 2016 , pp. 2498 - 2511
Copyright
© Cambridge University Press, 2015 

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