Drilling long geodesics in hyperbolic 3-manifolds

Given a complete hyperbolic 3-manifold one often wants to compare the original metric to a complete hyperbolic metric on the complement of some simple closed geodesic in the manifold. In some cases this can be done by interpolating between the two metrics using hyperbolic cone-manifolds. We refer to such a deformation as drilling and results which compare the geometry of the original manifold to the geometry of the drilled manifold as drilling theorems. The first results of this type are due to Hodgson and Kerckhoff ([HK2]). Their work was extended from finite volume manifolds to geometrically finite manifolds in [Br1]. In [BB] a strong version of the drilling theorem was proved that gave bi-Lipschitz control between the geometry of the two manifolds. In this paper we prove a drilling theorem that allows the geodesic to be arbitrarily long with the tradeoff that it must have a very large tubular neighborhood. We highlight two applications of this improved drilling theorem to classical conjectures about Kleinian groups. In [BS] the drilling theorem is applied to give a complete proof of the Bers-Sullivan-Thurston density conjecture. In [BBES] we give an alternate proof of the Brock-Canary-Minsky ending lamination classification ([Min], [BCM2], [BCM1]) which takes as its starting point Minsky’s a priori bounds theorem ([Min]). Note that there is also an approach to the density conjecture via the ending lamination classification. One can find a more complete history of these two conjectures in the papers cited above. We now give a more explicit description of the problem. Let (M, g) be a complete hyperbolic 3-manifold and γ a simple closed geodesic in M . Let M̂ = M\γ. Kerckhoff has observed that one can apply Thurston’s hyperbolization theorem to find a complete hyperbolic metric on M̂ (see [Ko]). If M is closed or finite volume then by Mostow-Prasad rigidity this metric will be unique. If (M, g) has infinite volume the metric will not be unique. To get a unique metric we need some extra structure. If (M, g) is geometrically finite then the higher genus ends of M are naturally compactified by (noded) Riemann