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On the pathwidth of hyperbolic 3-manifolds

Kristóf Huszár 1, 2
1 DATASHAPE - Understanding the Shape of Data
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
Abstract : According to Mostow's celebrated rigidity theorem, the geometry of closed hyperbolic 3-manifolds is already determined by their topology. In particular, the volume of such manifolds is a topological invariant and, as such, has been investigated for half a century. Motivated by the algorithmic study of 3-manifolds, Maria and Purcell have recently shown that every closed hyperbolic 3-manifold M with volume vol(M) admits a triangulation with dual graph of treewidth at most C vol(M), for some universal constant C. Here we improve on this result by showing that the volume provides a linear upper bound even on the pathwidth of the dual graph of some triangulation, which can potentially be much larger than the treewidth. Our proof relies on a synthesis of tools from 3-manifold theory: generalized Heegaard splittings, amalgamations, and the thick-thin decomposition of hyperbolic 3-manifolds. We provide an illustrated exposition of this toolbox and also discuss the algorithmic consequences of the result.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-03373577
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Submitted on : Monday, October 11, 2021 - 3:26:49 PM
Last modification on : Wednesday, October 13, 2021 - 3:38:00 AM

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  • HAL Id : hal-03373577, version 1
  • ARXIV : 2105.11371

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Kristóf Huszár. On the pathwidth of hyperbolic 3-manifolds. 2021. ⟨hal-03373577⟩

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