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

TL;DR

This paper establishes a bijective correspondence between 3D triangulations and certain hyperbolic handlebodies, revealing their geometric properties, and explores implications for hyperbolic 3-manifolds and Dehn fillings.

Contribution

It introduces a new bijective correspondence between triangulations and hyperbolic handlebodies, and studies their geometric and topological properties, including Dehn fillings.

Findings

Triangulations are uniquely determined by their 1-skeleton.

Manifolds with certain triangulations are hyperbolic with geodesic edges.

Every finite group is realizable as the isometry group of a hyperbolic 3-manifold.

Abstract

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.