Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 53C12, 53C23, 57R30, 58F15, 58F18, 53C22, 57M99, 58F25

Retrieve articles in all journals with MSC (1991): 53C12, 53C23, 57R30, 58F15, 58F18, 53C22, 57M99, 58F25


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
Email: fenley@math.princeton.edu, fenley@math.wustl.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-99-00304-5
PII: S 0894-0347(99)00304-5
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