Non-geometric veering triangulations

TL;DR

This paper introduces a computer program to generate veering ideal triangulations of mapping tori and presents the first known examples of non-geometric veering triangulations, including a minimal example with 13 tetrahedra.

Contribution

It develops Veering, a new software tool for constructing veering triangulations from surface homeomorphisms, and provides the first examples of non-geometric veering triangulations.

Findings

First examples of non-geometric veering triangulations.

Smallest non-geometric example has 13 tetrahedra.

Software automates triangulation generation from homeomorphisms.

Abstract

Recently, Ian Agol introduced a class of "veering" ideal triangulations for mapping tori of pseudo-Anosov homeomorphisms of surfaces punctured along the singular points. These triangulations have very special combinatorial properties, and Agol asked if these are "geometric", i.e. realised in the complete hyperbolic metric with all tetrahedra positively oriented. This paper describes a computer program Veering, building on the program Trains by Toby Hall, for generating these triangulations starting from a description of the homeomorphism as a product of Dehn twists. Using this we obtain the first examples of non-geometric veering triangulations; the smallest example we have found is a triangulation with 13 tetrahedra.