## Geometry & Topology

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

Danny Calegari

#### 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 $M̃$ can be partially compactified by a canonical cylinder $Suniv1×ℝ$ 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 $ℱ̃$. 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 $Homeo((Suniv1)t)$ 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

#### 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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