Local mixing and Abelian covers of compact hyperbolic manifolds

The virtually positive first Betti number conjecture (which is now a theorem) says that any compact hyperbolic 3-manifold admits an infinite abelian cover, after passing to a finite cover. We will discuss ergodic properties of the geodesic flow/frame flow on such abelian covers. In particular,
we will discuss recent joint work with Wenyu Pan on the local mixing property of the frame flow. We will also discuss applications to counting and equidistribution problems and to the measure classification problems.