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Mathematics > Geometric Topology

arXiv:1805.02357 (math)
[Submitted on 7 May 2018 (v1), last revised 22 Jan 2019 (this version, v2)]

Title:Treewidth, crushing, and hyperbolic volume

Authors:Clément Maria, Jessica S. Purcell
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Abstract:We prove that there exists a universal constant $c$ such that any closed hyperbolic 3-manifold admits a triangulation of treewidth at most $c$ times its 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.
Comments: 20 pages, 12 figures. V2: Section 4 has been rewritten, as the former argument (in V1) used a construction that relied on a wrong theorem. Section 5.1 has also been adjusted to the new construction. Various other arguments have been clarified
Subjects: Geometric Topology (math.GT); Combinatorics (math.CO)
MSC classes: 57M50, 57Q15, 57M15
Cite as: arXiv:1805.02357 [math.GT]
  (or arXiv:1805.02357v2 [math.GT] for this version)
  https://doi.org/10.48550/arXiv.1805.02357
Journal reference: Algebr. Geom. Topol. 19 (2019) 2625-2652
Related DOI: https://doi.org/10.2140/agt.2019.19.2625

Submission history

From: Jessica Purcell [view email]
[v1] Mon, 7 May 2018 06:17:36 UTC (128 KB)
[v2] Tue, 22 Jan 2019 01:20:41 UTC (114 KB)
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