Similar fillings and isolation of cusps of hyperbolic 3-manifolds

Roberto Frigerio

Vol. 229 (2007), No. 2, 339–364

DOI: 10.2140/pjm.2007.229.339

Abstract

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 prove that several elements in ℳ_g,k admit nonhomeomorphic 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.

Let N be a complete finite-volume hyperbolic 3-manifold with (possibly empty) geodesic boundary and cusps C₁,…,C_h,C_h+1,…,C_k. According to Neumann and Reid, the cusps C₁,…,C_h are said to be geometrically isolated from C_h+1,…,C_k if any small deformation of the hyperbolic structure on N induced by Dehn filling C_h+1,…,C_k does not affect the Euclidean structure at C₁,…,C_h. We show here that the cusps of any manifold in ℳ_g,k are geometrically isolated from each other. On the contrary, any element in ℳ_g,k admits an infinite family of hyperbolic Dehn fillings inducing nontrivial deformations of the hyperbolic structure on the geodesic boundary.

Keywords

Dehn filling, geodesic boundary, truncated tetrahedron, Kojima decomposition, commensurability

Mathematical Subject Classification 2000

Primary: 57M50

Secondary: 58H15

Milestones

Received: 15 July 2005

Accepted: 17 November 2006

Published: 1 February 2007