Applications of the drilling theorem
Abstract: We will show that the drilling theorem allows us to control parabolics in a sequence of geometrically finite hyperbolic 3-manifolds approximating a given hyperbolic 3-manifold. In the first application (joint work with J.Brock, K.Bromberg and J.Souto) we show that every point in the closure of the interior of an algebraic deformation space is approximated by a type-preserving sequence of geometrically finite hyperbolic 3-manifolds. We then conclude that, in almost all cases, the limit manifold is tame. An immediate corollary is Ahlfors' measure dichotomy for the closure of the interior of the algebraic deformation space. Our second application demonstrates a proof of the ending lamination conjecture for `super-slender' hyperbolic 3-manifolds. This proof does not require the construction of a model manifold but does require Minsky's work on end-invariants.