From pseudo-Anosov flows on graph manifolds to totally periodic flows

Abstract

We show that every pseudo-Anosov flow on a graph manifold is almost equivalent, i.e. orbit equivalent in the complement of a finite collection of closed orbits, to a totally periodic pseudo-Anosov flow or a suspension Anosov flow. The proof is via a hands-on construction of a partial Birkhoff section with genus one components that misses finitely many closed orbits. When combined with previous work of the author, this implies that every transitive Anosov flow on a graph manifold with orientable stable and unstable foliations is almost equivalent to a suspension Anosov flow.

Figures (10)

• Figure 1: Local picture of a pseudo-Anosov flow away from singular orbits (left) and near a 3-pronged singular orbit (right).
• Figure 2: Goodman-Fried surgery is performed by blowing up along a closed orbit $\gamma$ and blowing down along a different slope. It can be used to transform partial Birkhoff sections into partial cross sections.
• Figure 3: If two adjacent Birkhoff annuli in a quasi-transverse torus $T$ occupy adjacent quadrants at a boundary orbit, then $T$ can be homotoped to remove one tangent orbit.
• Figure 4: Local picture of a hyperbolic blow up of the geodesic flow.
• Figure 5: The prototypical round handle.

Theorems & Definitions (29)

• Conjecture 1.1: Fried Fri83, Christy Kir97, Ghys
• Theorem 1.2
• Corollary 1.3
• Example 2.1: Suspension Anosov flow
• Proposition 2.2: BM25
• Proposition 2.3
• proof
• Proposition 2.4: BF13
• Proposition 2.5: BF15
• Definition 2.6: Flows on 3-manifold with boundary