Similar fillings and isolation of cusps of hyperbolic 3-manifolds

Abstract

In this paper we deepen the analysis of certain classes M_{g,k} of hyperbolic 3-manifolds that were introduced in a previous work by B. Martelli, C. Petronio and the author. Each element of M_{g,k} is an oriented complete finite-volume hyperbolic 3-manifold with compact connected geodesic boundary of genus g and k cusps. We study small deformations of the complete hyperbolic structure of manifolds in M_{g,k} via a close analysis of their geodesic triangulations. We prove that several elements in M_{g,k} admit non-homeomorphic hyperbolic Dehn fillings sharing the same volume, homology, cusp volume, cusp shape, Heegaard genus, complex length of the shortest geodesic, length of the shortest return path, and Turaev-Viro invariants. Manifolds which share all these invariants are called geometrically similar, and were first studied by C. D. Hodgson, R. G. Meyerhoff and J. R. Weeks. The examples of geometrically similar manifolds they described are commensurable with each other. We show here that many elements in M_{g,k} admit non-commensurable geometrically similar Dehn fillings. The notion of geometric isolation for cusps in a hyperbolic 3-manifold was introduced by W. D. Neumann and A. W. Reid and studied by D. Calegary, who provided explanations for all the previously known examples of isolation phenomena. We show here that the cusps of any manifold M_{g,k} are geometrically isolated from each other. Apparently, isolation of cusps in our examples arises for different reasons from those described by Calegari. We also show that any element in M_{g,k} admits an infinite family of hyperbolic Dehn fillings inducing non-trivial deformations of the hyperbolic structure on the geodesic boundary.