Geometry & Topology, Vol. 4 (2000)
Paper no. 17, pages 457--515.
The Geometry of R-covered foliations
Danny Calegari
Abstract.
We study R-covered foliations of 3-manifolds from the point of view of
their transverse geometry. For an R-covered foliation in an atoroidal
3-manifold M, we show that M-tilde can be partially compactified by a
canonical cylinder S^1_univ x R on which pi_1(M) acts by elements of
Homeo(S^1) x Homeo(R), where the S^1 factor is canonically identified
with the circle at infinity of each leaf of F-tilde. We construct a
pair of very full genuine laminations transverse to each other and to
F, which bind every leaf of F. This pair of laminations can be blown
down to give a transverse regulating pseudo-Anosov flow for F,
analogous to Thurston's structure theorem for surface bundles over a
circle with pseudo-Anosov monodromy.
A corollary of the existence
of this structure is that the underlying manifold M is homotopy rigid
in the sense that a self-homeomorphism homotopic to the identity is
isotopic to the identity. Furthermore, the product structures at
infinity are rigid under deformations of the foliation F through
R-covered foliations, in the sense that the representations of pi_1(M)
in Homeo((S^1_univ)_t) are all conjugate for a family parameterized by
t. Another corollary is that the ambient manifold has word-hyperbolic
fundamental group.
Finally we speculate on connections between
these results and a program to prove the geometrization conjecture for
tautly foliated 3-manifolds.
Keywords.
Taut foliation, R-covered, genuine lamination, regulating flow, pseudo-Anosov, geometrization
AMS subject classification.
Primary: 57M50, 57R30.
Secondary: 53C12.
DOI: 10.2140/gt.2000.4.457
E-print: arXiv:math.GT/9903173
Submitted to GT on 18 September 1999.
(Revised 23 October 2000.)
Paper accepted 14 December 2000.
Paper published 14 December 2000.
Notes on file formats
Danny Calegari
Department of Mathematics, Harvard University
Cambridge, MA 02138, USA
Email: dannyc@math.harvard.edu
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