Treewidth, crushing and hyperbolic volume

Maria, Clément; Purcell, Jessica

doi:10.2140/agt.2019.19.2625

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Treewidth, crushing and hyperbolic volume

Clément Maria and Jessica S Purcell

Algebraic & Geometric Topology 19 (2019) 2625–2652

DOI: 10.2140/agt.2019.19.2625

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

References

Publication

Received: 8 August 2018

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

Volume 19, issue 5 (2019)