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
Abstract | References | Similar Articles | Additional Information
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.
-
dC99 D. Calegari, The geometry of $\mathbb {R}$-covered foliations I, math.GT/9903173.
- Danny Calegari, $\mathbf R$-covered foliations of hyperbolic $3$-manifolds, Geom. Topol. 3 (1999), 137–153. MR 1695533, DOI https://doi.org/10.2140/gt.1999.3.137 dC00 D. Calegari, Foliations with one-sided branching, preprint.
- Alberto Candel, Uniformization of surface laminations, Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 4, 489–516. MR 1235439
- 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, DOI https://doi.org/10.4310/mrl.1997.v4.n4.a14
- David Gabai and Ulrich Oertel, Essential laminations in $3$-manifolds, Ann. of Math. (2) 130 (1989), no. 1, 41–73. MR 1005607, DOI https://doi.org/10.2307/1971476
- Lucy Garnett, Foliations, the ergodic theorem and Brownian motion, J. Functional Analysis 51 (1983), no. 3, 285–311. MR 703080, DOI https://doi.org/10.1016/0022-1236%2883%2990015-0 lM00 L. Mosher, Laminations and flows transverse to finite depth foliations, Part I: Branched surfaces and dynamics, preprint.
- S. P. Novikov, The topology of foliations, Trudy Moskov. Mat. Obšč. 14 (1965), 248–278 (Russian). MR 0200938
- Dennis Sullivan, A homological characterization of foliations consisting of minimal surfaces, Comment. Math. Helv. 54 (1979), no. 2, 218–223. MR 535056, DOI https://doi.org/10.1007/BF02566269 wT97 W. Thurston, $3$-manifolds, foliations and circles I, math.GT/9712268. wT98 W. Thurston, $3$-manifolds, foliations and circles II, preprint. wT98b W. Thurston, Hyperbolic structures on $3$-manifolds II: Surface groups and $3$-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
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