Hyperbolic Cone-manifolds, Short Geodesics, and Schwarzian Derivatives

With his hyperbolic Dehn surgery theorem and later the orbifold theorem, Thurston demonstrated the power of using hyperbolic cone-manifolds to understand complete, non-singular hyperbolic 3-manifolds. Hodgson and Kerckhoff introduced analytic techniques to the study of cone-manifolds that they have used to prove deep results about finite volume hyperbolic 3-manifolds. In this paper we use Hodgson and Kerckhoff’s techniques to study infinite volume hyperbolic 3-manifolds. The results we will develop have many applications: the Bers density conjecture, the density of cusps on the boundary of quasiconformal deformations spaces, and for constructing type preserving sequences of Kleinian groups. The simplest example of the problem we will study is the following: Let M be a hyperbolic 3-manifold and c a simple closed geodesic in M . Then the topological manifold M\c also has a complete hyperbolic metric which we call M̂ . How does the geometry of M compare to that of M̂? Before attempting to answer such a question, we need to note that if M has infinite volume, the hyperbolic structure will not be unique. If we do not make further restrictions on the choice of M̂ , then there is no reason to expect that M and M̂ will be geometrically close. If M is convex co-compact, there is a natural choice to make for M̂ . Namely M is compactified by a conformal structure X . We then choose M̂ to be the unique geometrically finite hyperbolic structure on M\c with conformal boundary X . We can now return to our question: How do the geometry of M and M̂ compare? We will quantify this question in two ways. We will measure the length of geodesics inM and M̂ and we will measure the geometry of the ends ofM and M̂ by bounding the distance between the projective structures on their boundaries. What we will see is that the change in geometry is bounded by the length of the geodesic c in the original manifold M . Results of this type were first obtained by McMullen [Mc], in the case of a quasifuchsian manifold, where the geodesic c is also short on a component of the conformal boundary. This work has been extended to arbitrary geometrically finite manifolds by Canary, Culler, Hersonsky and Shalen [CCHS]. Their techniques are entirely different from ours and one goal of this paper is to give new proofs of their estimates. These estimates are a key step in proving the density of cusps in the boundary of quasiconformal deformation spaces of hyperbolic 3-manifolds.