  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.

Geometric Topology