Pseudo-Anosov flows in toroidal manifolds

TL;DR

This paper classifies pseudo-Anosov flows in toroidal 3-manifolds, proving rigidity in certain cases, describing their interaction with Seifert fibered pieces, and constructing new examples, including flows with one prong singularities.

Contribution

It provides new rigidity results for pseudo-Anosov flows in Seifert and solv manifolds, describes their structure in Seifert fibered pieces, and introduces a broad class of new, flexible examples.

Findings

Pseudo-Anosov flow in Seifert fibered manifolds is topologically equivalent to a geodesic flow.

Pseudo-Anosov flow in solv manifolds is equivalent to a suspension Anosov flow.

New classes of pseudo-Anosov flows with one prong singularities are constructed.

Abstract

We first prove rigidity results for pseudo-Anosov flows in prototypes of toroidal 3-manifolds: we show that a pseudo-Anosov flow in a Seifert fibered manifold is up to finite covers topologically equivalent to a geodesic flow and we show that a pseudo-Anosov flow in a solv manifold is topologically equivalent to a suspension Anosov flow. Then we study the interaction of a general pseudo-Anosov flow with possible Seifert fibered pieces in the torus decomposition: if the fiber is associated with a periodic orbit of the flow, we show that there is a standard and very simple form for the flow in the piece using Birkhoff annuli. This form is strongly connected with the topology of the Seifert piece. We also construct a large new class of examples in many graph manifolds, which is extremely general and flexible. We construct other new classes of examples, some of which are generalized pseudo-Anosov flows which have one prong singularities and which show that the above results in Seifert fibered and solvable manifolds do not apply to one prong pseudo-Anosov flows. Finally we also analyse immersed and embedded incompressible tori in optimal position with respect to a pseudo-Anosov flow.