Foliations with good geometry
Author:
Sérgio R. Fenley
Journal:
J. Amer. Math. Soc. 12 (1999), 619676
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 3manifolds 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 quasiisometric 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 tightthis is the best geometric behavior. We previously proved that nonsingular codimension one foliations in closed hyperbolic 3manifolds can never be quasiisometric. In this article we produce a large class of singular quasiisometric, codimension one foliations in closed hyperbolic 3manifolds. The foliations are stable and unstable foliations of pseudoAnosov flows. Our second result is an external result, relating (nonsingular) foliations in hyperbolic 3manifolds with their limit sets in the universal cover, that is, showing that leaves in the universal cover have good asymptotic behavior. Let be a Reebless, finite depth foliation in a closed hyperbolic 3manifold. Then is not quasiisometric, but suppose that is transverse to a quasigeodesic pseudoAnosov flow with quasiisometric stable and unstable foliationswhich are given by the internal result. We then show that the lifts of leaves of 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 pseudoAnosov flows and the topology of these foliations in the universal cover, and second a topological theory of the joint structure of the pseudoAnosov foliation in the universal cover.
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 S. Fenley, Surfaces transverse to pseudoAnosov flows and virtual fibers in manifolds, to appear in Topology (1999).
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Additional Information
Sérgio R. Fenley
Affiliation:
Department of Mathematics, Princeton University, Princeton, New Jersey 085441000
Address at time of publication:
Department of Mathematics, Washington University, St. Louis, Missouri 631304899
Email:
fenley@math.princeton.edu, fenley@math.wustl.edu
DOI:
http://dx.doi.org/10.1090/S0894034799003045
PII:
S 08940347(99)003045
Keywords:
Foliations,
flows,
hyperbolic 3manifolds,
geometric structures,
asymptotic geometry,
quasiisometries
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
