Treewidth, crushing and hyperbolic volume

Abstract

The treewidth of a $3$–manifold triangulation plays an important role in algorithmic $3$–manifold theory, and so it is useful to find bounds on the treewidth in terms of other properties of the manifold. We prove that there exists a universal constant $c$ such that any closed hyperbolic $3$–manifold admits a triangulation of treewidth at most the product of $c$ and the 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.