Treewidth, crushing and hyperbolic volume (Q2279065)

The treedwidth of a simplicial complex is a measure of complexity that requires some effort to define. As the authors put it, loosely speaking it is a ``measure of the sparsity of the gluing relations between tetrahedra''. Recently several algorithms have been found for 3-manifolds that are highly efficient for triangulations of small treewidth. The main theorem of the paper under review says that there exists a \(c>0\) such that any hyperbolic 3-manifold with volume \(\operatorname{vol}(M)\) admits a triangulation with treedwidth at most \(c\cdot \operatorname{vol}(M)\). As the authors explain, in practice the treewidth depends very much on the choice of a triangulation. In particular even if for theoretical reasons one knows that there exists a triangulation of low treedwidth, it might be very difficult to actually find it. Finally the authors also show that the converse to the main theorem does not hold. More precisely, they show that there exists a sequence of closed hyperbolic 3-manifolds with bounded treedwidth but with unbounded volumes.