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.

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Danny Calegari
Department of Mathematics, Harvard University
Cambridge, MA 02138, USA

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