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Geometry and topology of $\mathbb{R}$-covered foliations
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## Geometry and topology of $\mathbb{R}$-covered foliations

### Danny Calegari

**Abstract.**
An $\mathbb{R}$-covered foliation is a special type of taut foliation on a
$3$-manifold: one for which holonomy is defined for all transversals
and all time. The universal cover of a manifold $M$ with such a
foliation can be partially compactified by a cylinder at infinity,
somewhat analogous to the sphere at infinity of a hyperbolic manifold.
The action of $\pi_1(M)$ on this cylinder decomposes into a product
by elements of $\text{Homeo}(S^1)\times\text{Homeo}(\mathbb{R})$.
The action on the $S^1$ factor of this cylinder is rigid under deformations of the foliation through $\mathbb{R}$-covered foliations. Such a foliation admits
a pair of transverse genuine laminations whose complementary
regions are solid tori with finitely many boundary leaves, which
can be blown down to give a transverse regulating pseudo-Anosov
flow. These results all fit in an essential way into Thurston's
program to geometrize manifolds admitting taut foliations.

*Copyright 2000 American Mathematical Society*

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#### Article Info

- ERA Amer. Math. Soc.
**06** (2000), pp. 31-39
- Publisher Identifier: S 1079-6762(00)00077-9
- 2000
*Mathematics Subject Classification*. Primary 57M50
*Key words and phrases*.Foliations, laminations, $3$-manifolds, geometrization, $\mathbb{R}$-covered, product-covered, group actions on $\mathbb{R}$ and $S^1$
- Received by the editors May 7, 1999
- Posted on April 24, 2000
- Communicated by Walter Neumann
- Comments

**Danny Calegari**

Department of Mathematics, UC Berkeley, Berkeley, CA 94720

*E-mail address:* `dannyc@math.berkeley.edu`

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