Flows, growth rates, and the veering polynomial

Abstract

For certain pseudo-Anosov flows ϕ on closed 3-manifolds, unpublished work of Agol-Guéritaud produces a veering triangulation τ on the manifold M obtained by deleting ϕ's singular orbits. We show that τ can be realized in M so that its 2-skeleton is positively transverse to ϕ, and that the combinatorially defined flow graph Φ embedded in M uniformly codes ϕ's orbits in a precise sense. Together with these facts we use a modified version of the veering polynomial, previously introduced by the authors, to compute the growth rates of ϕ's closed orbits after cutting M along certain transverse surfaces, thereby generalizing work of McMullen in the fibered setting. These results are new even in the case where the transverse surface represents a class in the boundary of a fibered cone of M .Our work can be used to study the flow ϕ on the original closed manifold. Applications include counting growth rates of closed orbits after cutting along closed transverse surfaces, defining a continuous, convex entropy function on the 'positive' cone in H 1 of the cutopen manifold, and answering a question of Leininger about the closure of the set of all stretch factors arising as monodromies within a single fibered cone of a 3-manifold. This last application connects to the study of endperiodic automorphisms of infinite-type surfaces and the growth rates of their periodic points.

Computation of the Taut, the Veering and the Teichmüller Polynomials

“…It quickly became evident that veering triangulations exist also on non-fibered 3-manifolds [12,Section 4]. In unpublished work Agol and Guéritaud showed that a veering triangulation can be constructed from a pair (Ψ, Λ), where Ψ is a pseudo-Anosov flow on a closed 3-manifold, and Λ is a finite, nonempty collection of closed orbits of Ψ containing all singular orbits of Ψ and such that Ψ does not have perfect fits relative to Λ; see [14,Section 4]. The original fibered setup discussed in [1] is just a special case in which Ψ is the suspension flow of a pseudo-Anosov homeomorphism, and Λ consists only of the sigular orbits of Ψ.…”

“…The connection between veering triangulations and pseudo-Anosov flows has been explored further by Agol-Tsang [2], Landry-Minsky-Taylor [14], and Schleimer-Segerman [20,21]. From their work it follows that veering triangulations can be viewed as combinatorial tools to study pseudo-Anosov flows.…”

“…Of particular interest are slopes of boundary components of surfaces carried by V (if they exist). The manifold filled along such a slope admits an embedded surface almost transverse to Ψ(s)[14, Theorem 5.1].…”