Outdated Archival Version

These pages are not updated anymore. They reflect the state of 20 August 2005. For the current production of this journal, please refer to http://www.math.psu.edu/era/.

This journal is archived by the American Mathematical Society. The master copy is available at http://www.ams.org/era/

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

TeX source (38.4K)
DVI (40K)
PostScript (129.2K)

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