The geometry of ℝ–covered foliations

Abstract

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.