Regulating flows, topology of foliations and rigidity

Fenley, Sérgio

doi:10.1090/S0002-9947-05-03644-5

Regulating flows, topology of foliations and rigidity
HTML articles powered by AMS MathViewer

by Sérgio R. Fenley PDF

Trans. Amer. Math. Soc. 357 (2005), 4957-5000 Request permission

Abstract:

A flow transverse to a foliation is regulating if, in the universal cover, an arbitrary orbit of the flow intersects every leaf of the lifted foliation. This implies that the foliation is $\mathbf {R}$-covered, that is, its leaf space in the universal cover is homeomorphic to the reals. We analyse the converse of this implication to study the topology of the leaf space of certain foliations. We prove that if a pseudo-Anosov flow is transverse to an $\mathbf {R}$-covered foliation and the flow is not an $\mathbf {R}$-covered Anosov flow, then the flow is regulating for the foliation. Using this we show that several interesting classes of foliations are not $\mathbf {R}$-covered. Finally we show a rigidity result: if an $\mathbf {R}$-covered Anosov flow is transverse to a foliation but is not regulating, then the foliation blows down to one topologically conjugate to the stable or unstable foliations of the transverse flow.

References

D. V. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature. , Proceedings of the Steklov Institute of Mathematics, No. 90 (1967), American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by S. Feder. MR 0242194
D. V. Anosov and Ja. G. Sinaĭ, Certain smooth ergodic systems, Uspehi Mat. Nauk 22 (1967), no. 5 (137), 107–172 (Russian). MR 0224771
T. Barbot, Géométrie transverse des flots, Thesis, École Normale Sup. Lyon, 1992.
Thierry Barbot, Caractérisation des flots d’Anosov en dimension 3 par leurs feuilletages faibles, Ergodic Theory Dynam. Systems 15 (1995), no. 2, 247–270 (French, with English summary). MR 1332403, DOI 10.1017/S0143385700008361
Thierry Barbot, Flots d’Anosov sur les variétés graphées au sens de Waldhausen, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 5, 1451–1517 (French, with English and French summaries). MR 1427133
Thierry Barbot, Generalizations of the Bonatti-Langevin example of Anosov flow and their classification up to topological equivalence, Comm. Anal. Geom. 6 (1998), no. 4, 749–798. MR 1652255, DOI 10.4310/CAG.1998.v6.n4.a5
Andrew J. Casson and Steven A. Bleiler, Automorphisms of surfaces after Nielsen and Thurston, London Mathematical Society Student Texts, vol. 9, Cambridge University Press, Cambridge, 1988. MR 964685, DOI 10.1017/CBO9780511623912
Rufus Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math. 94 (1972), 1–30. MR 298700, DOI 10.2307/2373590
Rufus Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math. 95 (1973), 429–460. MR 339281, DOI 10.2307/2373793
Mark Brittenham, Essential laminations in Seifert-fibered spaces, Topology 32 (1993), no. 1, 61–85. MR 1204407, DOI 10.1016/0040-9383(93)90038-W
Danny Calegari, $\mathbf R$-covered foliations of hyperbolic $3$-manifolds, Geom. Topol. 3 (1999), 137–153. MR 1695533, DOI 10.2140/gt.1999.3.137
Danny Calegari, The geometry of $\textbf {R}$-covered foliations, Geom. Topol. 4 (2000), 457–515. MR 1800151, DOI 10.2140/gt.2000.4.457
Danny Calegari, Foliations with one-sided branching, Geom. Dedicata 96 (2003), 1–53. MR 1956833, DOI 10.1023/A:1022105922517
D. Calegari, Promoting essential laminations, Part I, preprint, 2001.
Alberto Candel, Uniformization of surface laminations, Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 4, 489–516. MR 1235439
M. Coornaert, T. Delzant, and A. Papadopoulos, Géométrie et théorie des groupes, Lecture Notes in Mathematics, vol. 1441, Springer-Verlag, Berlin, 1990 (French). Les groupes hyperboliques de Gromov. [Gromov hyperbolic groups]; With an English summary. MR 1075994
J. Cannon and W. Thurston, Group invariant Peano curves, preprint, 1985.
D. Cooper, D. D. Long, and A. W. Reid, Bundles and finite foliations, Invent. Math. 118 (1994), no. 2, 255–283. MR 1292113, DOI 10.1007/BF01231534
D. Cooper, D. D. Long, and A. W. Reid, Finite foliations and similarity interval exchange maps, Topology 36 (1997), no. 1, 209–227. MR 1410472, DOI 10.1016/0040-9383(95)00066-6
Travaux de Thurston sur les surfaces, Astérisque, vol. 66, Société Mathématique de France, Paris, 1979 (French). Séminaire Orsay; With an English summary. MR 568308
Sérgio R. Fenley, Asymptotic properties of depth one foliations in hyperbolic $3$-manifolds, J. Differential Geom. 36 (1992), no. 2, 269–313. MR 1180384
Sérgio R. Fenley, Quasi-isometric foliations, Topology 31 (1992), no. 3, 667–676. MR 1174265, DOI 10.1016/0040-9383(92)90057-O
Sérgio R. Fenley, Anosov flows in $3$-manifolds, Ann. of Math. (2) 139 (1994), no. 1, 79–115. MR 1259365, DOI 10.2307/2946628
Sérgio R. Fenley, Quasigeodesic Anosov flows and homotopic properties of flow lines, J. Differential Geom. 41 (1995), no. 2, 479–514. MR 1331975
Sérgio R. Fenley, The structure of branching in Anosov flows of $3$-manifolds, Comment. Math. Helv. 73 (1998), no. 2, 259–297. MR 1611703, DOI 10.1007/s000140050055
Sérgio R. Fenley, Continuous extension of Anosov foliations in $3$-manifolds with negatively curved fundamental group, Pacific J. Math. 186 (1998), no. 2, 201–216. MR 1663798, DOI 10.2140/pjm.1998.186.201
Sérgio R. Fenley, Limit sets of foliations in hyperbolic $3$-manifolds, Topology 37 (1998), no. 4, 875–894. MR 1607756, DOI 10.1016/S0040-9383(97)00062-1
Sérgio R. Fenley, Surfaces transverse to pseudo-Anosov flows and virtual fibers in $3$-manifolds, Topology 38 (1999), no. 4, 823–859. MR 1679801, DOI 10.1016/S0040-9383(98)00030-5
Sérgio R. Fenley, Foliations with good geometry, J. Amer. Math. Soc. 12 (1999), no. 3, 619–676. MR 1674739, DOI 10.1090/S0894-0347-99-00304-5
Sérgio R. Fenley, Foliations, topology and geometry of 3-manifolds: $\mathbf R$-covered foliations and transverse pseudo-Anosov flows, Comment. Math. Helv. 77 (2002), no. 3, 415–490. MR 1933786, DOI 10.1007/s00014-002-8348-9
Sérgio R. Fenley, Pseudo-Anosov flows and incompressible tori, Geom. Dedicata 99 (2003), 61–102. MR 1998929, DOI 10.1023/A:1024953221158
S. Fenley, Topological rigidity of pseudo-Anosov flows transverse to foliations, in preparation.
Sérgio Fenley and Lee Mosher, Quasigeodesic flows in hyperbolic 3-manifolds, Topology 40 (2001), no. 3, 503–537. MR 1838993, DOI 10.1016/S0040-9383(99)00072-5
John Franks and Bob Williams, Anomalous Anosov flows, Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979) Lecture Notes in Math., vol. 819, Springer, Berlin, 1980, pp. 158–174. MR 591182
David Gabai, Foliations and the topology of $3$-manifolds, J. Differential Geom. 18 (1983), no. 3, 445–503. MR 723813
David Gabai, Foliations and the topology of $3$-manifolds. II, J. Differential Geom. 26 (1987), no. 3, 461–478. MR 910017
David Gabai and Ulrich Oertel, Essential laminations in $3$-manifolds, Ann. of Math. (2) 130 (1989), no. 1, 41–73. MR 1005607, DOI 10.2307/1971476
Étienne Ghys, Flots d’Anosov sur les $3$-variétés fibrées en cercles, Ergodic Theory Dynam. Systems 4 (1984), no. 1, 67–80 (French, with English summary). MR 758894, DOI 10.1017/S0143385700002273
É. Ghys and P. de la Harpe (eds.), Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1990 (French). Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. MR 1086648, DOI 10.1007/978-1-4684-9167-8
Sue E. Goodman, Closed leaves in foliations of codimension one, Comment. Math. Helv. 50 (1975), no. 3, 383–388. MR 423371, DOI 10.1007/BF02565757
Sue Goodman and Sandi Shields, A condition for the stability of $\textbf {R}$-covered on foliations of $3$-manifolds, Trans. Amer. Math. Soc. 352 (2000), no. 9, 4051–4065. MR 1624178, DOI 10.1090/S0002-9947-00-02391-6
M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
André Haefliger, Groupoïdes d’holonomie et classifiants, Astérisque 116 (1984), 70–97 (French). Transversal structure of foliations (Toulouse, 1982). MR 755163
Michael Handel, Global shadowing of pseudo-Anosov homeomorphisms, Ergodic Theory Dynam. Systems 5 (1985), no. 3, 373–377. MR 805836, DOI 10.1017/S0143385700003011
John Hempel, $3$-Manifolds, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR 0415619
Nancy Kopell, Commuting diffeomorphisms, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 165–184. MR 0270396
Gilbert Levitt, Feuilletages des variétés de dimension $3$ qui sont des fibres en cercles, Comment. Math. Helv. 53 (1978), no. 4, 572–594 (French). MR 511848, DOI 10.1007/BF02566099
Brian S. Mangum, Incompressible surfaces and pseudo-Anosov flows, Topology Appl. 87 (1998), no. 1, 29–51. MR 1626072, DOI 10.1016/S0166-8641(97)00120-X
Lee Mosher, Dynamical systems and the homology norm of a $3$-manifold. I. Efficient intersection of surfaces and flows, Duke Math. J. 65 (1992), no. 3, 449–500. MR 1154179, DOI 10.1215/S0012-7094-92-06518-5
L. Mosher, Laminations and flows transverse to finite depth foliations, manuscript available in the web from http://newark.rutgers.edu:80/ mosher/, Part I: Branched surfaces and dynamics, Part II in preparation.
S. P. Novikov, The topology of foliations, Trudy Moskov. Mat. Obšč. 14 (1965), 248–278 (Russian). MR 0200938
J. F. Plante, Foliations with measure preserving holonomy, Ann. of Math. (2) 102 (1975), no. 2, 327–361. MR 391125, DOI 10.2307/1971034
J. F. Plante, Foliations of $3$-manifolds with solvable fundamental group, Invent. Math. 51 (1979), no. 3, 219–230. MR 530629, DOI 10.1007/BF01389915
J. F. Plante, Solvable groups acting on the line, Trans. Amer. Math. Soc. 278 (1983), no. 1, 401–414. MR 697084, DOI 10.1090/S0002-9947-1983-0697084-7
J. F. Plante, Anosov flows, transversely affine foliations, and a conjecture of Verjovsky, J. London Math. Soc. (2) 23 (1981), no. 2, 359–362. MR 609116, DOI 10.1112/jlms/s2-23.2.359
J. F. Plante and W. P. Thurston, Anosov flows and the fundamental group, Topology 11 (1972), 147–150. MR 295389, DOI 10.1016/0040-9383(72)90002-X
Rachel Roberts, Taut foliations in punctured surface bundles. I, Proc. London Math. Soc. (3) 82 (2001), no. 3, 747–768. MR 1816696, DOI 10.1112/plms/82.3.747
Rachel Roberts, Taut foliations in punctured surface bundles. II, Proc. London Math. Soc. (3) 83 (2001), no. 2, 443–471. MR 1839461, DOI 10.1112/plms/83.2.443
Dale Rolfsen, Knots and links, Mathematics Lecture Series, No. 7, Publish or Perish, Inc., Berkeley, Calif., 1976. MR 0515288
Harold Rosenberg, Foliations by planes, Topology 7 (1968), 131–138. MR 228011, DOI 10.1016/0040-9383(68)90021-9
Peter Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. (2) 17 (1978), no. 3, 555–565. MR 494062, DOI 10.1112/jlms/s2-17.3.555
Atle Selberg, On discontinuous groups in higher-dimensional symmetric spaces, Contributions to function theory (Internat. Colloq. Function Theory, Bombay, 1960) Tata Institute of Fundamental Research, Bombay, 1960, pp. 147–164. MR 0130324
Kurt Strebel, Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 5, Springer-Verlag, Berlin, 1984. MR 743423, DOI 10.1007/978-3-662-02414-0
Dennis Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math. 36 (1976), 225–255. MR 433464, DOI 10.1007/BF01390011
W. Thurston, Foliations of $3$-manifolds that are circle bundles, Ph.D. thesis, University of California Berkeley, 1972.
W. Thurston, The geometry and topology of 3-manifolds, Princeton University Lecture Notes, 1982.
William P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431. MR 956596, DOI 10.1090/S0273-0979-1988-15685-6
W. Thurston, Hyperbolic structures on $3$-manifolds II, Surface groups and $3$-manifolds that fiber over the circle, preprint.
W. Thurston, Three manifolds, foliations and circles I, preprint, 1997.
W. Thurston, Three manifolds, foliations and circles II, The transverse asymptotic geometry of foliations, preprint, 1998.
W. Thurston, Private communication.
Alberto Verjovsky, Codimension one Anosov flows, Bol. Soc. Mat. Mexicana (2) 19 (1974), no. 2, 49–77. MR 431281