Measurable Rigidity for Kleinian groups

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 $\hat{k}$ from the limit set ${\Lambda }_G$ 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 $\hat{k}$ always exists. In fact, we shall construct $\hat{k}$ concretely from the Cannon-Thurston maps of $G$ and $H$.

16 pages, no figures

References in corpus (1)

• Tameness of hyperbolic 3-manifolds