Mutations and faces of the Thurston norm ball dynamically represented by multiple distinct flows

Parlak, Anna¹

¹Department of Mathematics, University of California, Davis, Davis, CA, United States

Geometry & topology, Tome 29 (2025) no. 4, pp. 2105-2173

Résumé

A pseudo-Anosov flow on a hyperbolic 3-manifold dynamically represents a top-dimensional face F of the Thurston norm ball if the cone on F is dual to the cone spanned by the homology classes of closed orbits of the flow. Fried showed that for every fibered face of the Thurston norm ball there is a unique, up to isotopy and reparametrization, flow which dynamically represents the face. Using veering triangulations we have found that there are nonfibered faces of the Thurston norm ball which are dynamically represented by multiple topologically inequivalent flows. This raises the question of how distinct flows representing the same face are related.

We define combinatorial mutations of veering triangulations along surfaces that they carry. We give sufficient and necessary conditions for the mutant triangulation to be veering. After appropriate Dehn filling, these veering mutations correspond to transforming one 3-manifold M with a pseudo-Anosov flow transverse to an embedded surface S into another 3-manifold admitting a pseudo-Anosov flow transverse to a surface homeomorphic to S. We show that a nonfibered face of the Thurston norm ball can be dynamically represented by two distinct flows that differ by a veering mutation. Furthermore, one of the discussed pairs of homeomorphic veering mutants can be used to construct counterexamples to the classification theorem of Anosov flows on Bonatti–Langevin manifolds published in the 90s.

Détail

DOI : 10.2140/gt.2025.29.2105

Keywords: 3-manifolds, sutured manifolds, veering triangulations, pseudo-Anosov flows, mutations, Thurston norm

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