Foliations with One-Sided Branching

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 These laminations come from a universal circle, a refinement of the universal circles proposed by Thurston (unpublished), which maps monotonely and π₁(M)-equivariantly to each of the circles at infinity of the leaves of $\tilde F$, and is minimal with respect to this property.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.