  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.
