Algebraic & Geometric Topology

Treewidth, crushing and hyperbolic volume

Clément Maria and Jessica S Purcell

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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.

Article information

Source
Algebr. Geom. Topol., Volume 19, Number 5 (2019), 2625-2652.

Dates
Received: 8 August 2018
Revised: 21 January 2019
Accepted: 4 February 2019
First available in Project Euclid: 26 October 2019

Permanent link to this document
https://projecteuclid.org/euclid.agt/1572055269

Digital Object Identifier
doi:10.2140/agt.2019.19.2625

Mathematical Reviews number (MathSciNet)
MR4023324

Subjects
Primary: 57M15: Relations with graph theory [See also 05Cxx] 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M50: Geometric structures on low-dimensional manifolds

Keywords
$3$–manifold triangulation treewidth hyperbolic volume crushing normal surface

Citation

Maria, Clément; Purcell, Jessica S. Treewidth, crushing and hyperbolic volume. Algebr. Geom. Topol. 19 (2019), no. 5, 2625--2652. doi:10.2140/agt.2019.19.2625. https://projecteuclid.org/euclid.agt/1572055269


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