Pseudo-Anosov flows in toroidal manifolds

By Thierry Barbot, S. Fenley. 2010

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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. 37D20, 37D50; 57M60, 57R30 The goal of this article is to start a systematic study of pseudo-Anosov flows in toroidal 3‐manifolds. More specifically we analyse such flows in closed manifolds which are not hyperbolic or in pieces of the torus decomposition which are not hyperbolic and we obtain substantial results in Seifert fibered pieces. We also produce many new examples of pseudo-Anosov flows, including a large new class in graph manifolds and we study optimal position of tori with respect to arbitrary pseudo-Anosov flows. The study of hyperbolic flows in toroidal manifolds was initiated by Ghys [29], who analysed Anosov flows in 3‐dimensional circle bundles. Ghys showed that up to finite covers, the flow is topologically equivalent to the geodesic flow in the unit tangent bundle of a hyperbolic surface. This was later strengthened by the first author who showed that this holds if the manifold is Seifert fibered [2]. In the mid 70s

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/7eec655187346069db6a254f01fd6ba19b4cd743

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Citations in the corpus

Building Anosov flows on 3–manifolds (2014, background): “These examples also positively answer two questions asked by Barbot and Fenley in the final section of their recent paper [5].”