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Foliations with good geometry

Author: Sérgio R. Fenley
Journal: J. Amer. Math. Soc. 12 (1999), 619-676
MSC (1991): Primary 53C12, 53C23, 57R30, 58F15, 58F18; Secondary 53C22, 57M99, 58F25
Published electronically: April 26, 1999
MathSciNet review: 1674739
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Abstract: The goal of this article is to show that there is a large class of closed hyperbolic 3-manifolds admitting codimension one foliations with good large scale geometric properties. We obtain results in two directions. First there is an internal result: A possibly singular foliation in a manifold is quasi-isometric if, when lifted to the universal cover, distance along leaves is efficient up to a bounded multiplicative distortion in measuring distance in the universal cover. This means that leaves reflect very well the geometry in the large of the universal cover and are geometrically tight-this is the best geometric behavior. We previously proved that nonsingular codimension one foliations in closed hyperbolic 3-manifolds can never be quasi-isometric. In this article we produce a large class of singular quasi-isometric, codimension one foliations in closed hyperbolic 3-manifolds. The foliations are stable and unstable foliations of pseudo-Anosov flows. Our second result is an external result, relating (nonsingular) foliations in hyperbolic 3-manifolds with their limit sets in the universal cover, that is, showing that leaves in the universal cover have good asymptotic behavior. Let $\mathcal G$ be a Reebless, finite depth foliation in a closed hyperbolic 3-manifold. Then $\mathcal G$ is not quasi-isometric, but suppose that $\mathcal G$ is transverse to a quasigeodesic pseudo-Anosov flow with quasi-isometric stable and unstable foliations-which are given by the internal result. We then show that the lifts of leaves of $\mathcal G$ to the universal cover extend continuously to the sphere at infinity and we also produce infinitely many examples satisfying the hypothesis. The main tools used to prove these results are first a link between geometric properties of stable/unstable foliations of pseudo-Anosov flows and the topology of these foliations in the universal cover, and second a topological theory of the joint structure of the pseudo-Anosov foliation in the universal cover.

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Additional Information

Sérgio R. Fenley
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544-1000
Address at time of publication: Department of Mathematics, Washington University, St. Louis, Missouri 63130-4899

Keywords: Foliations, flows, hyperbolic 3-manifolds, geometric structures, asymptotic geometry, quasi-isometries
Received by editor(s): October 20, 1997
Received by editor(s) in revised form: March 5, 1998
Published electronically: April 26, 1999
Additional Notes: This research was partially supported by an NSF postdoctoral fellowship.
Article copyright: © Copyright 1999 American Mathematical Society