Pseudo-Anosov flows in toroidal manifolds

Barbot, Thierry¹ ; Fenley, Sérgio R²

¹ Avignon University, LMA, 33 Rue Louis Pasteur, 84000 Avignon, France
² Department of Mathematics, Florida State University, Room 208, 1017 Academic Way, Tallahassee, FL 32306-4510, USA

Geometry & topology, Tome 17 (2013) no. 4, pp. 1877-1954.

Résumé

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.

Détail

DOI : 10.2140/gt.2013.17.1877

Classification : 37D20, 37D50, 57M60, 57R30
Keywords: Pseudo-Anosov flows, toroidal manifolds, Seifert fibered spaces, graph manifolds

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