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Measurable rigidity for Kleinian groups
Published online by Cambridge University Press: 01 June 2015
Abstract
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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 
$H$ .
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