Classification of joinings for Kleinian groups

Abstract

We classify all locally finite joinings of a horospherical subgroup action on \Gamma \ G when \Gamma is a Zariski dense geometrically finite subgroup of G=PSL_2(R) or PSL_2(C). This generalizes Ratner's 1983 joining theorem for the case when \Gamma is a lattice in G. One of the main ingredients is equidistribution of non-closed horospherical orbits with respect to the Burger-Roblin measure which we prove in a greater generality where G is the connected component of the identity in SO(n,1) for n at least 2 and \Gamma is any Zariski dense geometrically finite subgroup of G.