Every pseudo-Anosov mapping class defines an associated veering triangulation of a punctured mapping torus. We show that generically, 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 Teichmüller theory, the Ending Lamination Theorem, study of the Thurston norm, and rigorous computation.
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David Futer, Samuel J. Taylor, William Worden, Random veering triangulations are not geometric. Groups Geom. Dyn. 14 (2020), no. 3, pp. 1077–1126DOI 10.4171/GGD/575