Triangulations of 3-manifolds, hyperbolic relative handlebodies, and Dehn filling

Details Triangulations of 3-manifolds, hyperbolic relative handlebodies, and Dehn filling Triangulations of 3-manifolds, hyperbolic relative handlebodies, and Dehn filling

2004-02-20

arxiv.org/abs/math/0402339v2

Comm. Math. Helv. 82 (2007), 903-934

Mathematics

Geometric Topology

Revised version, accepted for publication in Commentarii Mathematici Helvetici. 34 pages, 22 figures

Scientific paper

We establish a bijective correspondence between the set T(n) of 3-dimensional triangulations with n tetrahedra and a certain class H(n) of relative handlebodies (i.e. handlebodies with boundary loops, as defined by Johannson) of genus n+1. We show that the manifolds in H(n) are hyperbolic (with geodesic boundary, and cusps corresponding to the loops), have least possible volume, and simplest boundary loops. Mirroring the elements of H(n) in their geodesic boundary we obtain a class D(n) of cusped hyperbolic manifolds, previously considered by D. Thurston and the first named author. We show that also D(n) corresponds bijectively to T(n), and we study some Dehn fillings of the manifolds in D(n). As consequences of our constructions, we also show that: - A triangulation of a 3-manifold is uniquely determined up to isotopy by its 1-skeleton; - If a 3-manifold M has an ideal triangulation with edges of valence at least 6, then M is hyperbolic and the edges are homotopically non-trivial, whence homotopic to geodesics; - Every finite group G is the isometry group of a closed hyperbolic 3-manifold with volume less than a constant times |G|^9.

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