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$H$.