Triangulations of 3manifolds, hyperbolic relative handlebodies, and Dehn filling
Francesco Costantino
Université Paul Sabatier, Toulouse, FranceRoberto Frigerio
Università di Pisa, ItalyBruno Martelli
Università di Pisa, ItalyCarlo Petronio
Università di Pisa, Italy
Abstract
We establish a bijective correspondence between the set \mathcal{T}n of 3dimensional triangulations with n tetrahedra and a certain class ℋn of relative handlebodies (i.e. handlebodies with boundary loops, as defined by Johannson) of genus n + 1. We show that the manifolds in ℋn are hyperbolic (with geodesic boundary, and cusps corresponding to the loops), have least possible volume, and simplest boundary loops. Mirroring the elements of ℋn in their geodesic boundary we obtain a set \mathcal{D}n of cusped hyperbolic manifolds, previously considered by D. Thurston and the first named author. We show that also \mathcal{D}n corresponds bijectively to \mathcal{T}n, and we study some Dehn fillings of the manifolds in \mathcal{D}n. As consequences of our constructions, we also show that:

A triangulation of a 3manifold is uniquely determined up to isotopy by its 1skeleton;

If a 3manifold M has an ideal triangulation with edges of valence at least 6, then M is hyperbolic and the edges are homotopically nontrivial, whence homotopic to geodesics;

Every finite group G is the isometry group of a closed hyperbolic 3manifold with volume less than const × G9.