Transverse surfaces and pseudo-Anosov flows

TL;DR

This paper characterizes surfaces in 3-manifolds that are (almost) transverse to pseudo-Anosov flows, establishing a correspondence with veering triangulations and extending key theorems to boundary cases.

Contribution

It provides a characterization of almost transverse surfaces in pseudo-Anosov flows and links them to veering triangulations, answering an open question and extending existing theorems.

Findings

Surfaces almost transverse to pseudo-Anosov flows correspond to those carried by veering triangulations.

Any Thurston-norm minimizing surface with nonnegative pairing is almost transverse, up to isotopy.

Extension of Mosher's Transverse Surface Theorem to manifolds with boundary.

Abstract

Let $\varphi$ be a transitive pseudo-Anosov flow on an oriented, compact $3$-manifold $M$, possibly with toral boundary. We characterize the surfaces in $M$ that are (almost) transverse to $\phi$. When $\varphi$ has no perfect fits (e.g. $\varphi$ is the suspension flow of a pseudo-Anosov homeomorphism), we prove that any Thurston-norm minimizing surface $S$ that pairs nonnegatively with the closed orbits of $\varphi$ is almost transverse to $\varphi$, up to isotopy. This answers a question of Cooper--Long--Reid. Our main tool is a correspondence between surfaces that are almost transverse to $\varphi$ and those that are relatively carried by any associated veering triangulation. The correspondence also allows us to investigate the uniqueness of almost transverse position, to extend Mosher's Transverse Surface Theorem to the case with boundary, and more generally to characterize when relative homology classes represent Birkhoff surfaces.