Treewidth, crushing and hyperbolic volume

Clément Maria (1) , Jessica Purcell (2)

1 DATASHAPE - Understanding the Shape of Data
2 Monash University [Clayton]

The treewidth of a 3-manifold triangulation plays an important role in algorith-mic 3-manifold theory, and so it is useful to find bounds on the treewidth in terms of other properties of the manifold. In this paper, 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.

Domaines: Géométrie algorithmique [cs.CG]; Topologie géométrique [math.GT]

Clément Maria, Jessica Purcell. Treewidth, crushing and hyperbolic volume. Algebraic and Geometric Topology, 2019, 19 (5), pp.2625-2652. ⟨10.2140/agt.2019.19.2625⟩. ⟨hal-02429712⟩