Foliations with one-sided branching

Danny Calegari

Published 2001-01-03, updated 2002-09-17Version 4

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 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, which maps monotonely and group-equivariantly to each of the circles at infinity of the leaves of the universal cover of F, and is minimal with respect to this property. This circle is intimately bound up with the extrinsic geometry of the leaves of F. In particular, for any pair of leaves m,l of the universal cover of F with m>l, the leaf l is asymptotic to m in a dense set of directions at infinity, where the leaf space is co-oriented so that the foliation branches in the negative direction. This is a macroscopic version of an infinitesimal result of Thurston's and gives much more drastic control over the topology and geometry of F. The pair of laminations can be used to produce a pseudo-Anosov flow transverse to F which is regulating in the non-branching direction. Rigidity results for these laminations 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-covered foliation in math.GT/9903173, and these laminations become exactly the laminations constructed for the new branched foliation. Other corollaries include that the ambient manifold is d-hyperbolic in the sense of Gromov, and that a self-homeomorphism of this manifold homotopic to the identity is isotopic to the identity.