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Pseudo-Anosov Flows and Incompressible Tori

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Abstract

We study incompressible tori in 3-manifolds supporting pseudo-Anosov flows and more generally Z ⊕ Z subgroups of the fundamental group of such a manifold. If no element in this subgroup can be represented by a closed orbit of the pseudo-Anosov flow, we prove that the flow is topologically conjugate to a suspension of an Anosov diffeomorphism of the torus. In particular it is non singular and is an Anosov flow. It follows that either a pseudo-Anosov flow is topologically conjugate to a suspension Anosov flow, or any immersed incompressible torus can be realized as a free homotopy from a closed orbit of the flow to itself. The key tool is an analysis of group actions on non-Hausdorff trees, also known as R-order trees – we produce an invariant axis in the free action case. An application of these results is the following: suppose the manifold has an R-covered foliation transverse to a pseudo-Anosov flow. If the flow is not an R-covered Anosov flow, then it follows that the manifold is atoroidal.

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  1. Department of Mathematics, Florida State University, Tallahassee, FL, 32306, U.S.A.

    Sérgio R. Fenley

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Fenley, S.R. Pseudo-Anosov Flows and Incompressible Tori. Geometriae Dedicata 99, 61–102 (2003). https://doi.org/10.1023/A:1024953221158

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