Tameness on the boundary and Ahlfors' measure conjecture
A complete hyperbolic 3-manifold is said to be tame if it is homeomorphic to the interior of a compact 3-manifold. Marden's tameness conjecture is that a complete hyperbolic 3-manifold with finitely generated fundamental group is tame. In this paper the authors show that a complete hyperbolic 3-manifold \(N\) which is an algebraic limit of geometrically finite hyperbolic 3-manifolds is tame if \(N\) has non-empty conformal boundary. This result reduces the Ahlfors' measure conjecture to the following density conjecture: a complete hyperbolic 3-manifold with finitely generated fundamental group is an algebraic limit of geometrically finite hyperbolic 3-manifolds. The key theorem is that an algebraic limit of geometrically finite hyperbolic 3-manifolds is a limit of a type-preserving sequence of geometrically finite hyperbolic 3-manifolds. The authors also show the tameness with respect to a compression body and a strong limit of geometrically finite manifolds.
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