Random veering triangulations are not geometric

TL;DR

This paper demonstrates that most veering triangulations derived from pseudo-Anosov mapping classes are not geometric, using a combination of Teichmüller theory, lamination analysis, and computational methods.

Contribution

It establishes that generically, veering triangulations are non-geometric, providing a new understanding of their typical geometric properties in the context of mapping class groups.

Findings

Most veering triangulations are not geometric.

The non-geometric property holds for random walks and geodesic counts.

The proof combines theoretical and computational techniques.

Abstract

Every pseudo-Anosov mapping class $\varphi$ defines an associated veering triangulation $\tau_\varphi$ of a punctured mapping torus. We show that generically, $\tau_\varphi$ is not geometric. Here, the word "generic" can be taken either with respect to random walks in mapping class groups or with respect to counting geodesics in moduli space. Tools in the proof include Teichm\"uller theory, the Ending Lamination Theorem, study of the Thurston norm, and rigorous computation.