Geometry & Topology

Pseudo-Anosov flows in toroidal manifolds

Thierry Barbot and Sérgio R Fenley

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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.

Article information

Source
Geom. Topol., Volume 17, Number 4 (2013), 1877-1954.

Dates
Received: 19 November 2011
Revised: 22 March 2013
Accepted: 21 February 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732645

Digital Object Identifier
doi:10.2140/gt.2013.17.1877

Mathematical Reviews number (MathSciNet)
MR3109861

Zentralblatt MATH identifier
1317.37038

Subjects
Primary: 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37D50: Hyperbolic systems with singularities (billiards, etc.)
Secondary: 57M60: Group actions in low dimensions 57R30: Foliations; geometric theory

Keywords
Pseudo-Anosov flows toroidal manifolds Seifert fibered spaces graph manifolds

Citation

Barbot, Thierry; Fenley, Sérgio R. Pseudo-Anosov flows in toroidal manifolds. Geom. Topol. 17 (2013), no. 4, 1877--1954. doi:10.2140/gt.2013.17.1877. https://projecteuclid.org/euclid.gt/1513732645


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