Classification of joinings for Kleinian groups

Abstract

We classify all locally finite joinings of a horospherical subgroup action on $\Gamma \setminus G$ when $\Gamma$ is a Zariski-dense geometrically finite subgroup of $G={PSL}_2\left(\mathbb{R}\right)$ or ${PSL}_2\left(\mathbb{C}\right)$. This generalizes Ratner’s 1983joining theorem for the case when $\Gamma$ is a lattice in $G$. One of the main ingredients is equidistribution of nonclosed horospherical orbits with respect to the Burger–Roblin measure, which we prove in a greater generality where $\Gamma$ is any Zariski-dense geometrically finite subgroup of $G=SO(n,1{)}^{\circ }$, $n\ge 2$.