Geometry & Topology

The geometry of $\mathbb{R}$–covered foliations

Danny Calegari

Full-text: Open access

Abstract

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 M, we show that M̃ can be partially compactified by a canonical cylinder Suniv1× on which π1(M) acts by elements of Homeo(S1)× Homeo(), where the S1 factor is canonically identified with the circle at infinity of each leaf of ̃. We construct a pair of very full genuine laminations Λ± 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 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 through –covered foliations, in the sense that the representations of π1(M) in Homeo((Suniv1)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.

Article information

Source
Geom. Topol., Volume 4, Number 1 (2000), 457-515.

Dates
Received: 18 September 1999
Revised: 23 October 2000
Accepted: 14 December 2000
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513883294

Digital Object Identifier
doi:10.2140/gt.2000.4.457

Mathematical Reviews number (MathSciNet)
MR1800151

Zentralblatt MATH identifier
0964.57014

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds 57R30: Foliations; geometric theory
Secondary: 53C12: Foliations (differential geometric aspects) [See also 57R30, 57R32]

Keywords
taut foliation $\mathbb{R}$–covered genuine lamination regulating flow pseudo-Anosov geometrization

Citation

Calegari, Danny. The geometry of $\mathbb{R}$–covered foliations. Geom. Topol. 4 (2000), no. 1, 457--515. doi:10.2140/gt.2000.4.457. https://projecteuclid.org/euclid.gt/1513883294


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