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Regulating flows, topology of foliations and rigidity


Author: Sérgio R. Fenley
Journal: Trans. Amer. Math. Soc. 357 (2005), 4957-5000
MSC (2000): Primary 37D20, 53C12, 53C23, 57R30; Secondary 37C85, 57M99
DOI: https://doi.org/10.1090/S0002-9947-05-03644-5
Published electronically: March 10, 2005
MathSciNet review: 2165394
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Abstract: A flow transverse to a foliation is regulating if, in the universal cover, an arbitrary orbit of the flow intersects every leaf of the lifted foliation. This implies that the foliation is $\mathbf{R}$-covered, that is, its leaf space in the universal cover is homeomorphic to the reals. We analyse the converse of this implication to study the topology of the leaf space of certain foliations. We prove that if a pseudo-Anosov flow is transverse to an $\mathbf{R}$-covered foliation and the flow is not an $\mathbf{R}$-covered Anosov flow, then the flow is regulating for the foliation. Using this we show that several interesting classes of foliations are not $\mathbf{R}$-covered. Finally we show a rigidity result: if an $\mathbf{R}$-covered Anosov flow is transverse to a foliation but is not regulating, then the foliation blows down to one topologically conjugate to the stable or unstable foliations of the transverse flow.


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Additional Information

Sérgio R. Fenley
Affiliation: Department of Mathematics, Florida State University, Tallahassee, Florida 32306-4510

DOI: https://doi.org/10.1090/S0002-9947-05-03644-5
Received by editor(s): January 3, 2002
Received by editor(s) in revised form: February 1, 2004
Published electronically: March 10, 2005
Additional Notes: This research was partially supported by NSF grants DMS-9612317 and DMS-0071683.
Article copyright: © Copyright 2005 American Mathematical Society

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