A linear bound on the tetrahedral number of manifolds of bounded volume (after Jorgensen and Thurston)

TL;DR

This paper proves that the number of tetrahedra needed to triangulate certain hyperbolic 3-manifolds is linearly bounded by their volume, providing a topological interpretation of volume in terms of triangulation complexity.

Contribution

It offers a detailed proof of a folklore theorem relating manifold volume to tetrahedral triangulation bounds, with implications for link exteriors.

Findings

The d-neighborhood of the mu-thick part can be triangulated with a number of tetrahedra linearly bounded by volume.

The minimal tetrahedral triangulation of link exteriors is linearly proportional to the manifold's volume.

Provides a topological interpretation of hyperbolic volume via triangulation complexity.

Abstract

We provide a detailed proof of the following folklore theorem: Let mu > 0 be a Margulis constant for 3-dimensional hyperbolic space. Then for any d>0 there exists a constant K>0, depending on mu and d, so that for any complete finite volume hyperbolic 3-manifold M, the d-neighborhood of the mu-thick part of M can be triangulated using at most K Vol(M) tetrahedra; here Vol is the hyperbolic volume function. As a corollary, we obtain the following topological interpretation of the volume: the minimal number of tetrahedra required to triangulate a link exterior in M is linearly equivalent to Vol(M); for a precise statement see Corollary 1.3.