# zbMATH — the first resource for mathematics

Geometry and topology of $$\mathbb{R}$$-covered foliations. (English) Zbl 0995.57005
Summary: 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.

##### MSC:
 57M50 General geometric structures on low-dimensional manifolds 57R30 Foliations in differential topology; geometric theory 57N10 Topology of general $$3$$-manifolds (MSC2010) 57M60 Group actions on manifolds and cell complexes in low dimensions
Full Text:
##### References:
 [1] D. Calegari, The geometry of $$\mathbb R$$-covered foliations I, math.GT/9903173. · Zbl 0964.57014 [2] Danny Calegari, \?-covered foliations of hyperbolic 3-manifolds, Geom. Topol. 3 (1999), 137 – 153. · Zbl 0924.57014 [3] D. Calegari, Foliations with one-sided branching, preprint. · Zbl 1064.57015 [4] Alberto Candel, Uniformization of surface laminations, Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 4, 489 – 516. · Zbl 0785.57009 [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. · Zbl 0893.57012 [6] David Gabai and Ulrich Oertel, Essential laminations in 3-manifolds, Ann. of Math. (2) 130 (1989), no. 1, 41 – 73. · Zbl 0685.57007 [7] Lucy Garnett, Foliations, the ergodic theorem and Brownian motion, J. Funct. Anal. 51 (1983), no. 3, 285 – 311. · Zbl 0524.58026 [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). [10] Dennis Sullivan, A homological characterization of foliations consisting of minimal surfaces, Comment. Math. Helv. 54 (1979), no. 2, 218 – 223. · Zbl 0409.57025 [11] W. Thurston, $$3$$-manifolds, foliations and circles I, math.GT/9712268. [12] W. Thurston, $$3$$-manifolds, foliations and circles II, preprint. [13] W. Thurston, Hyperbolic structures on $$3$$-manifolds II: Surface groups and $$3$$-manifolds which fiber over the circle, math.GT/9801045.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.