Foliations with good geometry

Fenley, Sérgio^1, 2

¹Department of Mathematics, Princeton University, Princeton, New Jersey 08544-1000
²Department of Mathematics, Washington University, St. Louis, Missouri 63130-4899

Journal of the American Mathematical Society, Tome 12 (1999) no. 3, pp. 619-676

Résumé

The goal of this article is to show that there is a large class of closed hyperbolic 3-manifolds admitting codimension one foliations with good large scale geometric properties. We obtain results in two directions. First there is an internal result: A possibly singular foliation in a manifold is quasi-isometric if, when lifted to the universal cover, distance along leaves is efficient up to a bounded multiplicative distortion in measuring distance in the universal cover. This means that leaves reflect very well the geometry in the large of the universal cover and are geometrically tight—this is the best geometric behavior. We previously proved that nonsingular codimension one foliations in closed hyperbolic 3-manifolds can never be quasi-isometric. In this article we produce a large class of singular quasi-isometric, codimension one foliations in closed hyperbolic 3-manifolds. The foliations are stable and unstable foliations of pseudo-Anosov flows. Our second result is an external result, relating (nonsingular) foliations in hyperbolic 3-manifolds with their limit sets in the universal cover, that is, showing that leaves in the universal cover have good asymptotic behavior. Let $\mathcal G$ be a Reebless, finite depth foliation in a closed hyperbolic 3-manifold. Then $\mathcal G$ is not quasi-isometric, but suppose that $\mathcal G$ is transverse to a quasigeodesic pseudo-Anosov flow with quasi-isometric stable and unstable foliations—which are given by the internal result. We then show that the lifts of leaves of $\mathcal G$ to the universal cover extend continuously to the sphere at infinity and we also produce infinitely many examples satisfying the hypothesis. The main tools used to prove these results are first a link between geometric properties of stable/unstable foliations of pseudo-Anosov flows and the topology of these foliations in the universal cover, and second a topological theory of the joint structure of the pseudo-Anosov foliation in the universal cover.

Détail

DOI : 10.1090/S0894-0347-99-00304-5

Bibliographie

[1] Anosov, D. V. Geodesic flows on closed Riemann manifolds with negative curvature. 1969

[2] Anosov, D. V., Sinaĭ, Ja. G. Certain smooth ergodic systems Uspehi Mat. Nauk 1967 107 172

[3] Asimov, Daniel Round handles and non-singular Morse-Smale flows Ann. of Math. (2) 1975 41 54

[4] Barbot, Thierry Flots d’Anosov sur les variétés graphées au sens de Waldhausen Ann. Inst. Fourier (Grenoble) 1996 1451 1517

[5] Bestvina, Mladen, Mess, Geoffrey The boundary of negatively curved groups J. Amer. Math. Soc. 1991 469 481

[6] Casson, Andrew J., Bleiler, Steven A. Automorphisms of surfaces after Nielsen and Thurston 1988

[7] Bonahon, Francis Bouts des variétés hyperboliques de dimension 3 Ann. of Math. (2) 1986 71 158

[8] Brittenham, Mark Essential laminations in Seifert-fibered spaces Topology 1993 61 85

[9] Brittenham, Mark, Naimi, Ramin, Roberts, Rachel Graph manifolds and taut foliations J. Differential Geom. 1997 446 470

[10] Candel, Alberto Uniformization of surface laminations Ann. Sci. École Norm. Sup. (4) 1993 489 516

[11] Cantwell, J., Conlon, L. Smoothability of proper foliations Ann. Inst. Fourier (Grenoble) 1988 219 244

[12] Cooper, D., Long, D. D., Reid, A. W. Bundles and finite foliations Invent. Math. 1994 255 283

[13] Cooper, D., Long, D. D., Reid, A. W. Finite foliations and similarity interval exchange maps Topology 1997 209 227

[14] Fenley, Sérgio R. Asymptotic properties of depth one foliations in hyperbolic 3-manifolds J. Differential Geom. 1992 269 313

[15] Fenley, Sérgio R. Quasi-isometric foliations Topology 1992 667 676

[16] Fenley, Sérgio R. Anosov flows in 3-manifolds Ann. of Math. (2) 1994 79 115

[17] Fenley, Sérgio R. Quasigeodesic Anosov flows and homotopic properties of flow lines J. Differential Geom. 1995 479 514

[18] Fenley, Sérgio R. One sided branching in Anosov foliations Comment. Math. Helv. 1995 248 266

[19] Fenley, Sérgio R. The structure of branching in Anosov flows of 3-manifolds Comment. Math. Helv. 1998 259 297

[20] Franks, John, Williams, Bob Anomalous Anosov flows 1980 158 174

[21] Gabai, David Foliations and the topology of 3-manifolds J. Differential Geom. 1983 445 503

[22] Gabai, David Foliations and the topology of 3-manifolds. II J. Differential Geom. 1987 461 478

[23] Gabai, David Foliations and the topology of 3-manifolds. II J. Differential Geom. 1987 461 478

[24] Gabai, David, Oertel, Ulrich Essential laminations in 3-manifolds Ann. of Math. (2) 1989 41 73

[25] Sur les groupes hyperboliques d’après Mikhael Gromov 1990

[26] Gromov, M. Hyperbolic groups 1987 75 263

[27] Hempel, John 3-Manifolds 1976

[28] Jaco, William H., Shalen, Peter B. Seifert fibered spaces in 3-manifolds Mem. Amer. Math. Soc. 1979

[29] Levitt, Gilbert Foliations and laminations on hyperbolic surfaces Topology 1983 119 135

[30] Marden, Albert The geometry of finitely generated kleinian groups Ann. of Math. (2) 1974 383 462

[31] Mosher, Lee Dynamical systems and the homology norm of a 3-manifold. I. Efficient intersection of surfaces and flows Duke Math. J. 1992 449 500

[32] Mosher, Lee Dynamical systems and the homology norm of a 3-manifold. I. Efficient intersection of surfaces and flows Duke Math. J. 1992 449 500

[33] Mosher, Lee Examples of quasi-geodesic flows on hyperbolic 3-manifolds 1992 227 241

[34] Novikov, S. P. The topology of foliations Trudy Moskov. Mat. Obšč. 1965 248 278

[35] Palmeira, Carlos Frederico Borges Open manifolds foliated by planes Ann. of Math. (2) 1978 109 131

[36] Plante, J. F. Foliations with measure preserving holonomy Ann. of Math. (2) 1975 327 361

[37] Rosenberg, Harold Foliations by planes Topology 1968 131 138

[38] Strebel, Kurt Quadratic differentials 1984

[39] Scott, Peter The geometries of 3-manifolds Bull. London Math. Soc. 1983 401 487

[40] Sullivan, Dennis Cycles for the dynamical study of foliated manifolds and complex manifolds Invent. Math. 1976 225 255

[41] Thurston, William P. Three-dimensional manifolds, Kleinian groups and hyperbolic geometry Bull. Amer. Math. Soc. (N.S.) 1982 357 381

[42] Zeghib, A. Sur les feuilletages géodésiques continus des variétés hyperboliques Invent. Math. 1993 193 206