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Regulating flows, topology of foliations and rigidity

Author: Sérgio R. Fenley
Journal: Trans. Amer. Math. Soc. 357 (2005), 4957-5000
MSC (2000): Primary 37D20, 53C12, 53C23, 57R30; Secondary 37C85, 57M99
Published electronically: March 10, 2005
MathSciNet review: 2165394
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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.

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  • [An] D. V. Anosov, Geodesic flows on closed Riemannian manifolds with negative curvature, Proc. Steklov Inst. Math. 90 (1969). MR 0242194 (39:3527)
  • [An-Si] D. V. Anosov and Y. Sinai, Some smoothly ergodic systems, Russian Math. Surveys 22 (1967) 5 103-167.MR 0224771 (37:370)
  • [Ba1] T. Barbot, Géométrie transverse des flots, Thesis, École Normale Sup. Lyon, 1992.
  • [Ba2] T. Barbot, Caractérization des flots d'Anosov en dimension $3$ par leurs feuilletages faibles, Ergod. Th. Dynam. Sys. 15 (1995) 247-270.MR 1332403 (96d:58100)
  • [Ba3] T. Barbot, Flots d'Anosov sur les variétés graphées au sens de Waldhausen, Ann. Inst. Fourier Grenoble 46 (1996) 1451-1517.MR 1427133 (97j:57031)
  • [Ba4] T. Barbot, Generalizations of Bonatti-Langevin example of Anosov flow and their classification up to topological equivalence, Comm. Anal. Geom. 6 (1998) 749-798. MR 1652255 (99m:58142)
  • [Bl-Ca] S. Bleiler and A. Casson, Automorphisms of surfaces after Nielsen and Thurston, Cambridge Univ. Press, 1988. MR 0964685 (89k:57025)
  • [Bow1] R. Bowen, Periodic orbits for hyperbolic flows, Amer. Jour. Math. 94 (1970) 1-30. MR 0298700 (45:7749)
  • [Bow2] R. Bowen, Symbolic dynamics for hyperbolic flows, Amer. Jour. Math. 95 (1970) 429-459. MR 0339281 (49:4041)
  • [Br] M. Brittenham, Essential laminations in Seifert fibered spaces, Topology 32 (1993) 61-85. MR 1204407 (94c:57027)
  • [Ca1] D. Calegari, $\mathbf{R}$-covered foliations of hyperbolic $3$-manifolds, Geom. Topol 3 (1999) 137-153. MR 1695533 (2000c:57038)
  • [Ca2] D. Calegari, The geometry of $\mathbf{R}$-covered foliations, Geom. Topol. 4 (2000) 457-515. MR 1800151 (2001k:57016)
  • [Ca3] D. Calegari, Foliations with one sided branching, Geom. Dedicata 96 (2003) 1-53. MR 1956833 (2003m:57037)
  • [Ca4] D. Calegari, Promoting essential laminations, Part I, preprint, 2001.
  • [Can] A. Candel, Uniformization of surface laminations, Ann. Sci. Ecole Norm. Sup. (4) 26 (1993) 489-516.MR 1235439 (94f:57025)
  • [CDP] M. Coornaert, T. Delzant and A. Papadopoulos, Géométrie et théorie des groupes, Les groupes hyperboliques de Gromov, Lecture Notes in Math. 1441 Springer (1991). MR 1075994 (92f:57003)
  • [Ca-Th] J. Cannon and W. Thurston, Group invariant Peano curves, preprint, 1985.
  • [CLR1] D. Cooper, D. Long and A. Reid, Bundles and finite foliations, Inven. Math. 118 (1994) 255-283. MR 1292113 (96h:57013)
  • [CLR2] D. Cooper, D. Long and A. Reid, Finite foliations and similarity interval exchange maps, Topology, 36 (1997) 209-227. MR 1410472 (97j:57032)
  • [FLP] A. Fathi, Laudenbach, Poenaru, Travaux de Thurston sur les surfaces, Séminaire Orsay, Astérisque 66-67 Socieété Mathématique de France, 1979. MR 0568308 (82m:57003)
  • [Fe1] S. Fenley, Asymptotic properties of depth one foliations in hyperbolic $3$-manifolds, Jour. Diff. Geom., 36 (1992) 269-313.MR 1180384 (93k:57030)
  • [Fe2] S. Fenley, Quasi-isometric foliations, Topology 31 (1992) 667-676.MR 1174265 (94a:57044)
  • [Fe3] S. Fenley, Anosov flows in $3$-manifolds, Ann. of Math. 139 (1994) 79-115.MR 1259365 (94m:58162)
  • [Fe4] S. Fenley, Quasigeodesic Anosov flows and homotopic properties of closed orbits, Jour. Diff. Geo. 41 (1995) 479-514.MR 1331975 (96f:58118)
  • [Fe5] S. Fenley, The structure of branching in Anosov flows of $3$-manifolds, Comm. Math. Helv. 73 (1998) 259-297. MR 1611703 (99a:58123)
  • [Fe6] S. Fenley, Continuous extension of Anosov foliations in $3$-manifolds with negatively curved fundamental group, Pac. Jour. Math. 186 (1998) 201-216. MR 1663798 (99k:58135)
  • [Fe7] S. Fenley, Limit sets of foliations in hyperbolic $3$-manifolds, Topology 37 (1998) 875-894. MR 1607756 (2000e:57029)
  • [Fe8] S. Fenley, Surfaces transverse to pseudo-Anosov flows and virtual fibers in $3$-manifolds, Topology 38 (1999) 823-859. MR 1679801 (2001a:57041)
  • [Fe9] S. Fenley, Foliations with good geometry, Journal of the A.M.S. 12 (1999) 619-676. MR 1674739 (2000b:57041)
  • [Fe10] S. Fenley, Foliations and the topology of $3$-manifolds: $\mathbf{R}$-covered foliations and transverse pseudo-Anosov flows, Comm. Math. Helv. 77 (2002) 415-490. MR 1933786 (2003g:57044)
  • [Fe11] S. Fenley, Pseudo-Anosov flows and inconpressible tori, Geom. Dedicata 99 (2003) 61-102. MR 1998929 (2004f:57035)
  • [Fe12] S. Fenley, Topological rigidity of pseudo-Anosov flows transverse to foliations, in preparation.
  • [Fe-Mo] S. Fenley and L. Mosher, Quasigeodesic flows in hyperbolic $3$-manifolds, Topology 40 (2001) 503-537. MR 1838993 (2002e:57039)
  • [Fr-Wi] J. Franks and R. Williams, Anomalous Anosov flows, in Global theory of Dyn. Systems, Lecture Notes in Math. 819 Springer (1980). MR 0591182 (82e:58078)
  • [Ga1] D. Gabai, Foliations and the topology of 3-manifolds, J. Diff. Geo. 18 (1983) 445-503. MR 0723813 (86a:57009)
  • [Ga2] D. Gabai, Foliations and the topology of 3-manifolds III, J. Diff. Geo. 26 (1987) 479-536. MR 0910018 (89a:57014b)
  • [Ga-Oe] D. Gabai and U. Oertel, Essential laminations and $3$-manifolds, Ann. of Math. 130 (1989) 41-73.MR 1005607 (90h:57012)
  • [Gh] E. Ghys, Flots d'Anosov sur les $3$-variétés fibrées en cercles, Ergod. Th and Dynam. Sys. 4 (1984) 67-80.MR 0758894 (86b:58098)
  • [Gh-Ha] E. Ghys and P. de la Harpe, editors, Sur les groupes hyperboliques d'apres Mikhael Gromov, Progress in Math. 83 Birkhäuser, (1996).MR 1086648 (92f:53050)
  • [Go] S. Goodman, Closed leaves in foliations of codimension one, Comm. Math. Helv. 50 (1975) 383-388. MR 0423371 (54:11350)
  • [Go-Sh] S. Goodman and S. Shields, A condition for the stability of $\mathbf{R}$-covered for foliations of $3$-manifolds, Trans. A.M.S. 352 (2000) 4051-4065. MR 1624178 (2000m:57042)
  • [Gr] M. Gromov, Hyperbolic groups, in Essays on group theory, Springer, 1987, 75-263. MR 0919829 (89e:20070)
  • [Hae] A. Haefliger, Groupöides d'holonomie et classifiants, Asterisque 116 (1984) 70-97. MR 0755163 (86c:57026a)
  • [Han] M. Handel, Global shadowing of pseudo-Anosov homeomorphisms, Ergod. Th. Dynam. Sys. 5 (1985) 373-377. MR 0805836 (87e:58172)
  • [He] J. Hempel, 3-manifolds, Ann. of Math. Studies 86, Princeton University Press, 1976. MR 0415619 (54:3702)
  • [Ko] N. Kopell, Commuting diffeomorphisms, Proceedings of Symposia in Pure Mathematics, A.M.S. 14 (1970) 165-184. MR 0270396 (42:5285)
  • [Le] G. Levitt, Feuilletages des variétés de dimension trois qui son fibrés en cercles, Comm. Math. Helv. 53 (1978) 572-594. MR 0511848 (80c:57017)
  • [Man] B. Mangum, Incompressible surfaces and pseudo-Anosov flows, Topol. Appl. 87 (1998) 29-51. MR 1626072 (99e:57024)
  • [Mo1] L. Mosher, Dynamical systems and the homology norm of $3$-manifolds, I, Duke Jour. Math. 65 (1992) 449-500.MR 1154179 (93g:57018a)
  • [Mo2] L. Mosher, Laminations and flows transverse to finite depth foliations, manuscript available in the web from mosher/, Part I: Branched surfaces and dynamics, Part II in preparation.
  • [No] S. P. Novikov, Topology of foliations, Trans. Moscow Math. Soc. 14 (1963) 268-305. MR 0200938 (34:824)
  • [Pl1] J. Plante, Foliations with measure preserving holonomy, Ann. of Math. 107 (1975) 327-361. MR 0391125 (52:11947)
  • [Pl2] J. Plante, Foliations of $3$-manifolds with solvable fundamental group, Inven. Math. 51 (1979) 219-230. MR 0530629 (80i:57020)
  • [Pl3] J. Plante, Solvable groups acting on the line, Trans. A.M.S. 278 (1983) 401-414. MR 0697084 (85b:57048)
  • [Pl4] J. Plante, Anosov flows, transversely affine and a conjecture of Verjovsky, Jour. London Math. Soc. 83 (1981) 359-362. MR 0609116 (82g:58069)
  • [Pl-Th] J. Plante and W. Thurston, Anosov flows and the fundamental group, Topology 11 (1972) 147-150. MR 0295389 (45:4455)
  • [Rob1] R. Roberts, Taut foliations in punctured surface bundles, I, Proc. London Math. Soc. 82 (2001) 747-768.MR 1816696 (2003a:57040)
  • [Rob2] R. Roberts, Taut foliations in punctured surface bundles, II, Proc. London Math. Soc. 83 (2001) 443-471.MR 1839461 (2003j:57016)
  • [Rol] D. Rolfsen, Knots and links, Publish or Perish, 1976. MR 0515288 (58:24236)
  • [Ros] H. Rosenberg, Foliations by planes, Topology 7 (1968) 131-138. MR 0228011 (37:3595)
  • [Sc] P. Scott, Subgroups of surface groups are almost geometric, Jour. Lond. Math. Soc. 17 (1978) 555-565. MR 0494062 (58:12996)
  • [Se] A. Selberg, On discontinuous groups in higher-dimensional symmetric spaces, in Contributions to Function Theory, Tata Inst. of Fund. Research, Bombay (1960) 147-164. MR 0130324 (24:A188)
  • [St] K. Strebel, Quadratic differentials, Springer-Verlag, 1984. MR 0743423 (86a:30072)
  • [Su] D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math. 36 (1976) 225-255. MR 0433464 (55:6440)
  • [Th1] W. Thurston, Foliations of $3$-manifolds that are circle bundles, Ph.D. thesis, University of California Berkeley, 1972.
  • [Th2] W. Thurston, The geometry and topology of 3-manifolds, Princeton University Lecture Notes, 1982.
  • [Th3] W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. A.M.S 19 (1988) 417-431. MR 0956596 (89k:57023)
  • [Th4] W. Thurston, Hyperbolic structures on $3$-manifolds II, Surface groups and $3$-manifolds that fiber over the circle, preprint.
  • [Th5] W. Thurston, Three manifolds, foliations and circles I, preprint, 1997.
  • [Th6] W. Thurston, Three manifolds, foliations and circles II, The transverse asymptotic geometry of foliations, preprint, 1998.
  • [Th7] W. Thurston, Private communication.
  • [Ve] A. Verjovsky, Codimension one Anosov flows, Bol. Soc. Mat. Mex. 19 (1977) 49-77.MR 0431281 (55:4282)

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Additional Information

Sérgio R. Fenley
Affiliation: Department of Mathematics, Florida State University, Tallahassee, Florida 32306-4510

Received by editor(s): January 3, 2002
Received by editor(s) in revised form: February 1, 2004
Published electronically: March 10, 2005
Additional Notes: This research was partially supported by NSF grants DMS-9612317 and DMS-0071683.
Article copyright: © Copyright 2005 American Mathematical Society

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