On the pathwidth of hyperbolic 3-manifolds

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DOI:

https://doi.org/10.57717/cgt.v1i1.4

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 subject of investigations 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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Published

2022-02-02

How to Cite

Huszár, K. (2022). On the pathwidth of hyperbolic 3-manifolds. Computing in Geometry and Topology, 1(1), 1:1–1:19. https://doi.org/10.57717/cgt.v1i1.4

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Original Research Articles

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