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January 2009 Geometry of foliations and flows I: Almost transverse pseudo-Anosov flows and asymptotic behavior of foliations
Sérgio R. Fenley
J. Differential Geom. 81(1): 1-89 (January 2009). DOI: 10.4310/jdg/1228400628

Abstract

Let $\mathcal{F}$ be a foliation in a closed 3-manifold with negatively curved fundamental group and suppose that $\mathcalF$ is almost transverse to a quasigeodesic pseudo-Anosov flow. We show that the leaves of the foliation in the universal cover extend continuously to the sphere at infinity; therefore the limit sets of the leaves are continuous images of the circle. One important corollary is that if $\mathcal{F}$ is a Reebless, infinite depth foliation in a hyperbolic 3-manifold, then it has the continuous extension property. Such infinite depth foliations exist whenever the second Betti number is non zero. The result also applies to other classes of foliations, including a large class of foliations where all leaves are dense, and infinitely many examples with one sided branching. One extremely useful tool is a detailed understanding of the topological structure and asymptotic properties of the 1-dimensional foliations in the leaves of e $\mathcal{F}$ induced by the stable and unstable foliations of the flow.

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Sérgio R. Fenley. "Geometry of foliations and flows I: Almost transverse pseudo-Anosov flows and asymptotic behavior of foliations." J. Differential Geom. 81 (1) 1 - 89, January 2009. https://doi.org/10.4310/jdg/1228400628

Information

Published: January 2009
First available in Project Euclid: 4 December 2008

zbMATH: 1160.57026
MathSciNet: MR2477891
Digital Object Identifier: 10.4310/jdg/1228400628

Rights: Copyright © 2009 Lehigh University

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Vol.81 • No. 1 • January 2009
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