On the pathwidth of hyperbolic 3-manifolds

TL;DR

This paper proves that the volume of hyperbolic 3-manifolds linearly bounds the pathwidth of their triangulation dual graphs, advancing understanding of their topological complexity and algorithmic properties.

Contribution

It improves previous bounds by establishing a linear relationship between volume and pathwidth, using advanced 3-manifold theory tools.

Findings

Volume linearly bounds the pathwidth of triangulation dual graphs.

The proof synthesizes Heegaard splittings, amalgamations, and thick-thin decomposition.

Discusses algorithmic implications of the new bounds.

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.