Geometry and topology of -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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

An -covered foliation is a special type of taut foliation on a -manifold: one for which holonomy is defined for all transversals and all time. The universal cover of a manifold 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 on this cylinder decomposes into a product by elements of . The action on the factor of this cylinder is rigid under deformations of the foliation through -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.

**1.**D. Calegari,*The geometry of -covered foliations I*, math.GT/9903173.**2.**Danny Calegari,*𝐑-covered foliations of hyperbolic 3-manifolds*, Geom. Topol.**3**(1999), 137–153 (electronic). MR**1695533**, 10.2140/gt.1999.3.137**3.**D. Calegari,*Foliations with one-sided branching*, preprint.**4.**Alberto Candel,*Uniformization of surface laminations*, Ann. Sci. École Norm. Sup. (4)**26**(1993), no. 4, 489–516. MR**1235439****5.**David Gabai and William H. Kazez,*Homotopy, isotopy and genuine laminations of 3-manifolds*, Geometric topology (Athens, GA, 1993) AMS/IP Stud. Adv. Math., vol. 2, Amer. Math. Soc., Providence, RI, 1997, pp. 123–138. MR**1470725****6.**David Gabai and Ulrich Oertel,*Essential laminations in 3-manifolds*, Ann. of Math. (2)**130**(1989), no. 1, 41–73. MR**1005607**, 10.2307/1971476**7.**Lucy Garnett,*Foliations, the ergodic theorem and Brownian motion*, J. Funct. Anal.**51**(1983), no. 3, 285–311. MR**703080**, 10.1016/0022-1236(83)90015-0**8.**L. Mosher,*Laminations and flows transverse to finite depth foliations, Part I: Branched surfaces and dynamics*, preprint.**9.**S. P. Novikov,*The topology of foliations*, Trudy Moskov. Mat. Obšč.**14**(1965), 248–278 (Russian). MR**0200938****10.**Dennis Sullivan,*A homological characterization of foliations consisting of minimal surfaces*, Comment. Math. Helv.**54**(1979), no. 2, 218–223. MR**535056**, 10.1007/BF02566269**11.**W. Thurston,*-manifolds, foliations and circles I*, math.GT/9712268.**12.**W. Thurston,*-manifolds, foliations and circles II*, preprint.**13.**W. Thurston,*Hyperbolic structures on -manifolds II: Surface groups and -manifolds which fiber over the circle*, math.GT/9801045.

Retrieve articles in *Electronic Research Announcements of the American Mathematical Society*
with MSC (2000):
57M50

Retrieve articles in all journals with MSC (2000): 57M50

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

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