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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
Maria, Clément 1 ; Purcell, Jessica 2
@article{10_2140_agt_2019_19_2625,
author = {Maria, Cl\'ement and Purcell, Jessica},
title = {Treewidth, crushing and hyperbolic volume},
journal = {Algebraic and Geometric Topology},
pages = {2625--2652},
publisher = {mathdoc},
volume = {19},
number = {5},
year = {2019},
doi = {10.2140/agt.2019.19.2625},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.2625/}
}
TY - JOUR AU - Maria, Clément AU - Purcell, Jessica TI - Treewidth, crushing and hyperbolic volume JO - Algebraic and Geometric Topology PY - 2019 SP - 2625 EP - 2652 VL - 19 IS - 5 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.2625/ DO - 10.2140/agt.2019.19.2625 ID - 10_2140_agt_2019_19_2625 ER -
%0 Journal Article %A Maria, Clément %A Purcell, Jessica %T Treewidth, crushing and hyperbolic volume %J Algebraic and Geometric Topology %D 2019 %P 2625-2652 %V 19 %N 5 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.2625/ %R 10.2140/agt.2019.19.2625 %F 10_2140_agt_2019_19_2625
Maria, Clément; Purcell, Jessica. Treewidth, crushing and hyperbolic volume. Algebraic and Geometric Topology, Tome 19 (2019) no. 5, pp. 2625-2652. doi: 10.2140/agt.2019.19.2625
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