Tameness on the boundary and Ahlfors’ measure conjecture

Abstract

Let N be a complete hyperbolic 3-manifold that is an algebraic limit of geometrically finite hyperbolic 3-manifolds. We show N is homeomorphic to the interior of a compact 3-manifold, or tame, if one of the following conditions holds: 1. N has non-empty conformal boundary, 2. N is not homotopy equivalent to a compression body, or 3. N is a strong limit of geometrically finite manifolds. The first case proves Ahlfors’ measure conjecture for Kleinian groups in the closure of the geometrically finite locus: given any algebraic limit Γ of geometrically finite Kleinian groups, the limit set of Γ is either of Lebesgue measure zero or all of Ĉ. Thus, Ahlfors’ conjecture is reduced to the density conjecture of Bers, Sullivan, and Thurston.