Injectivity radius bounds in hyperbolic convex cores.

Abstract

In this dissertation, we investigate the geometry of convex cores of hyperbolic 3-manifolds. Specifically, we show that if M is a book of I-bundles or an acylindrical, hyperbolizable 3-manifold, then there exists a uniform upper bound on injectivity radius for points in the convex core of any hyperbolic 3-manifold homeomorphic to the interior of M. The main theorem relies on our extension of a theorem of Steve Kerckhoff and William Thurston. Their theorem established the existence of an upper bound on injectivity radius for points in the convex core of hyperbolic 3-manifolds without cusps where the manifolds are homotopy equivalent to a closed surface; the extension includes the possibility of cusps. The main theorem also partially answers a conjecture of Curt McMullen which hypothesized the existence of an upper bound on the radius of balls that can be embedded in the convex core of hyperbolic 3-manifolds with finitely generated fundamental group, where the bound depends on the number of generators of the fundamental group. A corollary concerns the behavior of a sequence of limit sets associated to a sequence of Kleinian groups. Consider a sequence of Kleinian groups isomorphic to the fundamental group of either a book of I-bundles or an acylindrical, hyperbolizable 3-manifold. If the sequence converges geometrically to a nonabelian group, then their associated limit sets converge in the Hausdorff topology. The dissertation begins with a discussion of the problem, the main results, and applications of the main theorem. Background material on hyperbolic geometry and Kleinian groups is presented in Chapter Three. Chapters Four and Five provide an introduction to and extension of the theory of simplicial hyperbolic surfaces and continuous families of simplicial hyperbolic surfaces first developed by Richard Canary. Chapter Six is an extension of the Kerckhoff-Thurston result; Chapter Seven then uses this result to prove the main theorem in the case that M is a book of I-bundles. Chapter Eight discusses the case the acylindrical case, and the final chapter discusses some consequences of the main theorem.