Geometry of foliations and flows i:almost transverse pseudo-anosov flows and asymptotic behavior of foliations

Sergio R. Fenley Florida State University

Differential Geometry mathscidoc:1609.10089

j. differential geometry, 81, (1), 1-89, 2009
Let F be a foliation in a closed 3-manifold with negatively curved fundamental group and suppose that F is almost trans- verse to a quasigeodesic pseudo-Anosov °ow. We show that the leaves of the foliation in the universal cover extend continuously to the sphere at in¯nity; therefore the limit sets of the leaves are continuous images of the circle. One important corollary is that if F is a Reebless, ¯nite depth foliation in a hyperbolic 3-manifold, then it has the continuous extension property. Such ¯nite 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 in¯nitely 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 F induced by the stable and unstable foliations of the °ow.
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@inproceedings{sergio2009geometry,
  title={GEOMETRY OF FOLIATIONS AND FLOWS I:ALMOST TRANSVERSE PSEUDO-ANOSOV FLOWS AND ASYMPTOTIC BEHAVIOR OF FOLIATIONS},
  author={Sergio R. Fenley},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160909142821646027746},
  booktitle={j. differential geometry},
  volume={81},
  number={1},
  pages={1-89},
  year={2009},
}
Sergio R. Fenley. GEOMETRY OF FOLIATIONS AND FLOWS I:ALMOST TRANSVERSE PSEUDO-ANOSOV FLOWS AND ASYMPTOTIC BEHAVIOR OF FOLIATIONS. 2009. Vol. 81. In j. differential geometry. pp.1-89. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160909142821646027746.
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