Veering Triangulations: Theory and Experiment

Abstract

Certain fibered hyperbolic 3-manifolds admit a layered veering triangulation, which can be constructed algorithmically given the stable lamination of the monodromy. These triangulations were introduced by Agol in 2011, and have been further studied by several others in the years since. In the first part of this work, we obtain experimental results which shed light on the combinatorial structure of veering triangulations, and its relation to certain topological invariants of the underlying manifold. Among other things, our experimental results strongly suggest that typical veering triangulations are non-geometric, i.e., they cannot be realized as a union of isometrically embedded hyperbolic tetrahedra. In the second part, we prove that veering triangulations are in fact generically non-geometric.