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The Geometry of R-covered foliations

Calegari, Danny (2000) The Geometry of R-covered foliations. Geometry and Topology, 4 (17). pp. 457-515. ISSN 1465-3060. doi:10.2140/gt.2000.4.457.

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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.

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Additional Information:Submitted to GT on 18 September 1999. (Revised 23 October 2000.) Paper accepted 14 December 2000. Paper published 14 December 2000. Proposed: David Gabai; Seconded: Dieter Kotschick, Walter Neumann I would like to thank Andrew Casson, Sergio Fenley and Bill Thurston for their invaluable comments, criticisms and inspiration. A cursory glance at the list of references will indicate my indebtedness to Bill for both general and specific guidance throughout this project. I would also like to thank John Stallings and Benson Farb for helping me out with some remedial group theory. In addition, I am extremely grateful to the referee for providing numerous valuable comments and suggestions, which have tremendously improved the clarity and the rigour of this paper. I would also like to point out that I had some very useful conversations with Sergio after part of this work was completed. Working independently, he went on to find proofs of many of the results in the last section of this paper, by somewhat different methods. In particular, he found a construction of the laminations [Lambda][plus-minus sign] by using the theory of earthquakes as developed by Thurston.
Subject Keywords:Taut foliation, R-covered, genuine lamination, regulating flow, pseudo-Anosov, geometrization
Issue or Number:17
Record Number:CaltechAUTHORS:CALgt00
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:1193
Deposited By: Archive Administrator
Deposited On:04 Jan 2006
Last Modified:08 Nov 2021 19:08

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