#### Volume 4, issue 1 (2000)

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

### Danny Calegari

Geometry & Topology 4 (2000) 457–515
 arXiv: math.GT/9903173
##### 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 $\stackrel{̃}{M}$ can be partially compactified by a canonical cylinder ${S}_{univ}^{1}×ℝ$ on which ${\pi }_{1}\left(M\right)$ acts by elements of $Homeo\left({S}^{1}\right)×Homeo\left(ℝ\right)$, where the ${S}^{1}$ factor is canonically identified with the circle at infinity of each leaf of $\stackrel{̃}{\mathsc{ℱ}}$. We construct a pair of very full genuine laminations ${\Lambda }^{±}$ transverse to each other and to $\mathsc{ℱ}$, which bind every leaf of $\mathsc{ℱ}$. This pair of laminations can be blown down to give a transverse regulating pseudo-Anosov flow for $\mathsc{ℱ}$, 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 $\mathsc{ℱ}$ through $ℝ$–covered foliations, in the sense that the representations of ${\pi }_{1}\left(M\right)$ in $Homeo\left({\left({S}_{univ}^{1}\right)}_{t}\right)$ 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, $\mathbb{R}$–covered, genuine lamination, regulating flow, pseudo-Anosov, geometrization
##### Mathematical Subject Classification 2000
Primary: 57M50, 57R30
Secondary: 53C12
##### 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
 Danny Calegari Department of Mathematics Harvard University Cambridge Massachusetts 02138 USA