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Pseudo-Anosov flows in toroidal manifolds

Thierry Barbot and Sérgio R Fenley

Geometry & Topology 17 (2013) 1877–1954

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.

Pseudo-Anosov flows, toroidal manifolds, Seifert fibered spaces, graph manifolds
Mathematical Subject Classification 2010
Primary: 37D20, 37D50
Secondary: 57M60, 57R30
Received: 19 November 2011
Revised: 22 March 2013
Accepted: 21 February 2013
Published: 10 July 2013
Proposed: Danny Calegari
Seconded: Walter Neumann, Leonid Polterovich
Thierry Barbot
Avignon University
33 Rue Louis Pasteur
84000 Avignon
Sérgio R Fenley
Department of Mathematics
Florida State University
Room 208, 1017 Academic Way
Tallahassee, FL 32306-4510