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Treewidth, crushing and hyperbolic volume

Clément Maria and Jessica S Purcell

Algebraic & Geometric Topology 19 (2019) 2625–2652

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.

$3$–manifold triangulation, treewidth, hyperbolic volume, crushing normal surface
Mathematical Subject Classification 2010
Primary: 57M15, 57M25, 57M50
Received: 8 August 2018
Revised: 21 January 2019
Accepted: 4 February 2019
Published: 20 October 2019
Clément Maria
INRIA Sophia Antipolis-Méditerranée
Jessica S Purcell
School of Mathematics
Monash University
Monash University, VIC