Examples of Anosov flows with genus one Birkhoff sections

Abstract

We show that a transitive Anosov flow with orientable stable and unstable foliations that either (i) admits a Birkhoff section whose first return map is a Penner type pseudo-Anosov map, or (ii) is totally periodic admits a genus one Birkhoff section. This provides evidence for a conjecture of Fried and Ghys. The proof utilizes a result of the author on the horizontal Goodman surgery operation. To apply this result for showing (i), we establish correspondence between horizontal Goodman surgery on pseudo-Anosov flows and horizontal surgery on veering triangulations in the layered setting.

Figures (53)

• Figure 1: Left: A local picture of an Anosov flow. Right: A local picture of its orbit space.
• Figure 2: Given an Anosov-type fully-punctured pseudo-Anosov map $f:S^\circ \to S^\circ$, we can construct an Anosov flow by collapsing the extension $\overline{\phi}^t_f$ of the suspension flow along boundary tori. The image of $S^\circ$ becomes the interior of a Birkhoff section.
• Figure 3: A local picture of a pseudo-Anosov flow near a singular orbit.
• Figure 4: A local example of a positive curve (in red) and a negative curve (blue), projected along the flow direction onto a plane.
• Figure 5: Left/right: Local form of a positive/negative surgery curve, respectively.

Theorems & Definitions (112)

• Conjecture 1.1: Fried, Ghys
• Theorem 1.2: Dehornoy-Shannon DS19
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5: Tsa24
• Theorem 1.6
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4