Geometry & Topology 4 (2000) 457–515

DOI: 10.2140/gt.2000.4.457

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 can be partially compactified by a canonical cylinder S¹_univ×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((S¹_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.

Keywords

taut foliation, R–covered, genuine lamination, regulating flow, pseudo-Anosov, geometrization

Mathematical Subject Classification

Primary: 57M50, 57R30

Secondary: 53C12

References

Publication

Received: 18 September 1999
Revised: 23 October 2000
Accepted: 14 December 2000
Published: 14 December 2000
Proposed: David Gabai
Seconded: Dieter Kotschick, Walter Neumann

Authors