Ends of hyperbolic 3-manifolds

Abstract

Let N = H 3 / Γ N = {{\mathbf {H}}^3}/\Gamma be a hyperbolic 3 3 -manifold which is homeomorphic to the interior of a compact 3 3 -manifold. We prove that N N is geometrically tame. As a consequence, we prove that Γ \Gamma ’s limit set L Γ {L_\Gamma } is either the entire sphere at infinity or has measure zero. We also prove that N N ’s geodesic flow is ergodic if and only if L Γ {L_\Gamma } is the entire sphere at infinity.