On the pathwidth of hyperbolic 3-manifolds

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. * This paper is based on previously unpublished parts of the author's PhD thesis [24]. † Supported by the French government through the 3IA Côte d'Azur Investments in the Future project managed by the National Research Agency (ANR) under the reference number ANR-19-P3IA-0002.1 The treewidth is a structural graph parameter measuring the "tree-likeness" of a graph, cf. Section 3.2 For related work on FPT-algorithms in knot theory, see [10] and [35] and the references therein.3 The running times are measured in terms of the number of tetrahedra in the input triangulation.4 Some of these algorithms [14,16] have been implemented in the topology software Regina [8,11].