Geometriae Dedicata

, Volume 96, Issue 1, pp 1–53

Foliations with One-Sided Branching

  • Danny Calegari
Article

DOI: 10.1023/A:1022105922517

Cite this article as:
Calegari, D. Geometriae Dedicata (2003) 96: 1. doi:10.1023/A:1022105922517

Abstract

We generalize the main results from the author's paper in Geom. Topol. 4 (2000), 457–515 and from Thurston's eprint math.GT/9712268 to taut foliations with one-sided branching. First constructed by Meigniez, these foliations occupy an intermediate position between ℝ-covered foliations and arbitrary taut foliations of 3-manifolds. We show that for a taut foliation \(F\) with one-sided branching of an atoroidal 3-manifold M, one can construct a pair of genuine laminations Λ± of M transverse to \(F\) with solid torus complementary regions which bind every leaf of \(F\) in a geodesic lamination. These laminations come from a universal circle, a refinement of the universal circles proposed by Thurston (unpublished), which maps monotonely and π1(M)-equivariantly to each of the circles at infinity of the leaves of \(\tilde F\), and is minimal with respect to this property. This circle is intimately bound up with the extrinsic geometry of the leaves of \(\tilde F\). In particular, let \(\tilde F\) denote the pulled-back foliation of the universal cover, and co-orient \(\tilde F\) so that the leaf space branches in the negative direction. Then for any pair of leaves of \(\tilde F\) with μλ, the leaf λ is asymptotic to μ in a dense set of directions at infinity. This is a macroscopic version of an infinitesimal result from Thurston and gives much more drastic control over the topology and geometry of \(F\), than is achieved by him. The pair of laminations Λ± can be used to produce a pseudo-Anosov flow transverse to \(F\) which is regulating in the nonbranching direction. Rigidity results for Λ± in the ℝ-covered case are extended to the case of one-sided branching. In particular, an ℝ-covered foliation can only be deformed to a foliation with one-sided branching along one of the two laminations canonically associated to the ℝ-coveredfoliation constructed in Geom. Topol. 4 (2000), 457–515, and these laminations become exactly the laminations Λ± for the new branched foliation. Other corollaries include that the ambient manifold is δ-hyperbolic in the sense of Gromov, and that a self-homeomorphism of this manifold homotopic to the identity is isotopic to the identity.

foliation lamination one-sided branching 3-manifold geometrization pseudo-Anosov 

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Danny Calegari
    • 1
  1. 1.Department of MathematicsHarvard UniversityCambridgeU.S.A.

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