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

Abstract

We establish a bijective correspondence between the set ${\mathcal{T}}_n$ of 3-dimensional triangulations with $n$ tetrahedra and a certain class ${\mathcal{H}}_n$ of relative handlebodies (i.e. handlebodies with boundary loops, as defined by Johannson) of genus $n+1$.

We show that the manifolds in ${\mathcal{H}}_n$ are hyperbolic (with geodesic boundary, and cusps corresponding to the loops), have least possible volume, and simplest boundary loops.

Mirroring the elements of ${\mathcal{H}}_n$ 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 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 $\mathrm{const}\times |G{|}^9$.