Transverse surfaces and pseudo-Anosov flows

TL;DR

The paper provides a complete framework for understanding when surfaces in a 3-manifold with a transitive pseudo-Anosov flow can be made almost transverse (and when truly transverse) by exploiting veering triangulations. It establishes a precise equivalence: a surface is almost transverse to φ if and only if it is relatively carried by an associated veering triangulation, and this correspondence yields a robust Transverse Surface Theorem, extends Mosher’s theorem to manifolds with boundary, and clarifies when relative homology classes are carried by Birkhoff-type surfaces. The work also develops dynamic blowups and shadow theory to prove uniqueness of transverse position and to realize boundary phenomena, including the construction of Birkhoff surfaces and boundary-periodic leaves. Overall, the methods fuse geometric topology, combinatorial veering machinery, and flow dynamics to connect Thurston norm geometry with surface transversality in pseudo-Anosov flows, even in the presence of boundary. The results yield new tools for identifying transverse representatives, understanding their uniqueness, and representing Thurston-norm faces via dynamics.

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 $φ$. 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.

Figures (23)

• Figure 1: The standard veering tetrahedron. Each face is cooriented out of the page.
• Figure 2: A local picture of part of an edge of a veering triangulation, showing the 2-skeleton's branched surface structure.
• Figure 3: Left: a truncated veering tetrahedron. Right: the tips of truncated tetrahedra divide the boundary of $M_U$ into upward and downward ladders (green and blue, respectively) separated by ladderpole curves.
• Figure 4: An example of a cusped solid torus. To obtain a cusped torus shell, we would drill out a neighborhood of the core circle.
• Figure 5: The stable and unstable branched surfaces of a veering triangulation intersect veering tetrahedra as shown, where $L$ and $R$ labels denote left and right veering edges, respectively. Note the veers of the bottom edges in the top row are irrelevant, as are the veers of the top edges in the bottom row.

Theorems & Definitions (91)

• Theorem A: Strong transverse surface theorem
• Theorem B
• Theorem C: Representing relative classes with Birkhoff surfaces
• Theorem D
• Theorem E: Almost transverse if and only if relatively carried
• Lemma 2.1
• proof
• Proposition 2.2
• Lemma 2.3
• proof