Local mixing and invariant measures for horospherical subgroups on abelian covers

Abelian covers of hyperbolic $3$-manifolds are ubiquitous. We prove the local mixing theorem of the frame flow for abelian covers of closed hyperbolic $3$-manifolds. We obtain a classification theorem for measures invariant under the horospherical subgroup. We also describe applications to the prime geodesic theorem as well as to other counting and equidistribution problems. Our results are proved for any abelian cover of a homogeneous space $\Gamma_0\backslash G$ where $G$ is a rank one simple Lie group and $\Gamma_0<G$ is a convex cocompact Zariski dense subgroup.