Tameness on the boundary and Ahlfors' measure conjecture
  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 G of geometrically finite Kleinian groups, the limit set of G is either of Lebesgue measure zero or all of the Riemann sphere. Thus, Ahlfors' conjecture is reduced to the density conjecture of Bers, Sullivan, and Thurston. 
Jeffrey Brock, Kenneth Bromberg, Richard Evans and Juan Souto