Free Seifert pieces of pseudo-Anosov flows

TL;DR

This paper characterizes the structure of pseudo-Anosov flows on free Seifert fibered pieces of 3-manifolds, showing they are orbitally equivalent to hyperbolic blow ups of geodesic flow pieces, expanding understanding of flow dynamics.

Contribution

It introduces a detailed structure theorem for free Seifert pieces, including the concept of hyperbolic blow ups and almost k-convergence groups, advancing the classification of pseudo-Anosov flows.

Findings

Pseudo-Anosov flows on free Seifert pieces are orbitally equivalent to hyperbolic blow ups of geodesic flows.

Introduction of almost k-convergence groups and a new convergence theorem.

Development of an alternative model for geodesic flows on hyperbolic surfaces.

Abstract

We prove a structure theorem for pseudo-Anosov flows restricted to Seifert fibered pieces of three manifolds. The piece is called periodic if there is a Seifert fibration so that a regular fiber is freely homotopic, up to powers, to a closed orbit of the flow. A non periodic Seifert fibered piece is called free. In a previous paper [Ba-Fe1] we described the structure of a pseudo-Anosov flow restricted to a periodic piece up to isotopy along the flow. In the present paper we consider free Seifert pieces. We show that, in a carefully defined neighborhood of the free piece, the pseudo-Anosov flow is orbitally equivalent to a hyperbolic blow up of a geodesic flow piece. A geodesic flow piece is a finite cover of the geodesic flow on a compact hyperbolic surface, usually with boundary. In the proof we introduce almost k-convergence groups and prove a convergence theorem. We also introduce an alternative model for the geodesic flow of a hyperbolic surface that is suitable to prove these results, and we carefully define what is a hyperbolic blow up.