We study –covered
foliations of 3–manifolds from the point of view of their transverse geometry. For an
–covered foliation in an
atoroidal 3–manifold ,
we show that
can be partially compactified by a canonical cylinder
on which
acts by
elements of ,
where the
factor is canonically identified with the circle at infinity of each leaf of
.
We construct a pair of very full genuine laminations
transverse to each
other and to , which
bind every leaf of .
This pair of laminations can be blown down to give a transverse regulating pseudo-Anosov
flow for ,
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
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
through
–covered
foliations, in the sense that the representations of
in
are all conjugate for a
family parameterized by .
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.
