#### Volume 19, issue 5 (2019)

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Treewidth, crushing and hyperbolic volume

### Clément Maria and Jessica S Purcell

Algebraic & Geometric Topology 19 (2019) 2625–2652
##### 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.

##### Keywords
$3$–manifold triangulation, treewidth, hyperbolic volume, crushing normal surface
##### Mathematical Subject Classification 2010
Primary: 57M15, 57M25, 57M50
##### Publication
Revised: 21 January 2019
Accepted: 4 February 2019
Published: 20 October 2019
##### Authors
 Clément Maria INRIA Sophia Antipolis-Méditerranée Valbonne France Jessica S Purcell School of Mathematics Monash University Monash University, VIC Australia