Measurable rigidity for Kleinian groups

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 $H$ .