Treewidth, crushing, and hyperbolic volume

TL;DR

This paper establishes a linear relationship between the volume of hyperbolic 3-manifolds and their triangulation treewidth, while also showing that bounded treewidth does not imply bounded volume, with implications for 3-manifold topology.

Contribution

It proves a universal bound relating hyperbolic volume and triangulation treewidth and demonstrates that bounded treewidth does not constrain volume growth, introducing new techniques involving normal surface crushing.

Findings

Existence of a universal constant c relating volume and treewidth.

Bounded treewidth does not imply bounded volume in hyperbolic 3-manifolds.

Crushing normal surfaces does not increase carving-width, affecting treewidth by a constant.

Abstract

We prove that there exists a universal constant $c$ such that any closed hyperbolic 3-manifold admits a triangulation of treewidth at most $c$ times its volume. The converse is not true: we show there exists a sequence of hyperbolic 3-manifolds of bounded treewidth but volume approaching infinity. Along the way, we prove that crushing a normal surface in a triangulation does not increase the carving-width, and hence crushing any number of normal surfaces in a triangulation affects treewidth by at most a constant multiple.