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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