Measurable Rigidity for Kleinian groups

Let $G,H$ be two Kleinian groups with homeomorphic quotients ${\mathbb{H}}^3\mathrm{/}G$ and ${\mathbb{H}}^3\mathrm{/}H$. 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 ${\mathrm{\Lambda }}_G$ of $G$ to that of $H$ is either the restriction of a M"{o}bius 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 $H$.