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

Author:
Danny Calegari

Journal:
Electron. Res. Announc. Amer. Math. Soc. **6** (2000), 31-39

MSC (2000):
Primary 57M50

DOI:
https://doi.org/10.1090/S1079-6762-00-00077-9

Published electronically:
April 24, 2000

MathSciNet review:
1756133

Full-text PDF Free Access

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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.

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Additional Information

**Danny Calegari**

Affiliation:
Department of Mathematics, UC Berkeley, Berkeley, CA 94720

MR Author ID:
605373

Email:
dannyc@math.berkeley.edu

Keywords:
Foliations,
laminations,
$3$-manifolds,
geometrization,
$\mathbb {R}$-covered,
product-covered,
group actions on $\mathbb {R}$ and $S^1$

Received by editor(s):
May 7, 1999

Published electronically:
April 24, 2000

Communicated by:
Walter Neumann

Article copyright:
© Copyright 2000
American Mathematical Society