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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
Published electronically: April 24, 2000
MathSciNet review: 1756133
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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
Email: dannyc@math.berkeley.edu

DOI: http://dx.doi.org/10.1090/S1079-6762-00-00077-9
PII: S 1079-6762(00)00077-9
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