#### Volume 17, issue 4 (2013)

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Pseudo-Anosov flows in toroidal manifolds

### Thierry Barbot and Sérgio R Fenley

Geometry & Topology 17 (2013) 1877–1954
##### 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.

##### Keywords
Pseudo-Anosov flows, toroidal manifolds, Seifert fibered spaces, graph manifolds
##### Mathematical Subject Classification 2010
Primary: 37D20, 37D50
Secondary: 57M60, 57R30