Mutations and Geometric Invariants of Hyperbolic 3-Manifolds

Abstract

The main goal of this thesis is to examine the quality of geometric invariants of finite volume hyperbolic 3-manifolds. In particular, we examine how to construct large classes of hyperbolic 3-manifolds that are geometrically similar: they have a number of geometric invariants that are the same, but are non-isometric. Large classes of geometrically similar hyperbolic 3-manifolds provide examples where the minimal geometric data needed to determine M must be quite large. For our constructions, we will use a cut and paste operation known as mutation. Ruberman has shown that mutations of hyperelliptic surfaces inside hyperbolic 3-manifolds preserve volume. Here, we provide geometric and topological conditions under which such mutations also preserve the initial length spectrum. This work requires us to analyze when least area surfaces could intersect short geodesics in a hyperbolic 3-manifold. As a corollary of this result, we show that the number of hyperbolic knot complements with the same volume and the same initial length spectrum grows at least factorially fast with the volume and the number of twist regions; a similar statement holds for closed hyperbolic 3-manifolds, obtained via Dehn surgery. Furthermore, we show that the knot complements used for this construction are pairwise incommensurable by analyzing their cusp shapes.