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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 can be partially compactified by a canonical
cylinder S1univ×R on which
π1(M) acts
by elements of Homeo(S1)×Homeo(R),
where the S1 factor is canonically identified with the circle at
infinity of each leaf of ~F. We construct a pair
of very full genuine laminations Λ±
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 π1(M) in
Homeo((S1univ)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.
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