Skip to main content
Log in

Promoting essential laminations

  • Published:
Inventiones mathematicae Aims and scope

Abstract

We show that a co-orientable taut foliation of a closed, orientable, algebraically atoroidal 3-manifold is either the weak stable foliation of an Anosov flow, or else there are a pair of very full genuine laminations transverse to the foliation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Alperin, R. (ed.): Arboreal Group Theory. MSRI Publications, vol. 19. New York: Springer 1991

  2. Anosov, D.: Geodesic flows on closed Riemannian manifolds with negative curvature. Proc. Steklov Inst. Math. 90 (1969). Translated from the Russian, Providence, RI: American Mathematical Society 1969

  3. Barbot, T.: Caractérisation des flots d’Anosov en dimension 3 par leurs feuilletages faibles. Ergodic Theory Dyn. Syst. 15, 247–270 (1995)

    MATH  MathSciNet  Google Scholar 

  4. Bing, R.H.: The geometric topology of 3-manifolds. AMS Colloquium Publications 40, (1983). Providence, RI: American Mathematical Society 1983

  5. Calegari, D.: The geometry of \(\mathbb{R}\)-covered foliations. Geom. Topol. 4, 457–515 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  6. Calegari, D.: Foliations with one-sided branching. Geom. Dedicata 96, 1–53 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Calegari, D.: Leafwise smoothing laminations. Algebr. Geom. Topol. 1, 579–585 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  8. Calegari, D., Dunfield, N.: Laminations and groups of homeomorphisms of the circle. Invent. Math. 152, 149–204 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  9. Calegari, D.: Dynamical forcing of circular group. Trans. Am. Math. Soc. 358, 3473–3491 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Calegari, D.: Circular groups, planar groups and the Euler class. Geom. Topol. Monogr. 7, 431–491 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  11. Calegari, D.: Universal circles for quasigeodesic flows. eprint math.GT/0406040

  12. Candel, A.: Uniformization of surface laminations. Ann. Sci. Éc. Norm. Supér., IV. Sér. 26, 489–516 (1993)

    Google Scholar 

  13. Candel, A.: The harmonic measures of Lucy Garnett. Adv. Math. 176, 187–247 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Candel, A., Conlon, L.: Foliations I. AMS Graduate Studies in Mathematics, vol. 23. Providence, RI: American Mathematical Society 2000

  15. Candel, A., Conlon, L.: Foliations II. AMS Graduate Studies in Mathematics, vol. 60. Providence, RI: American Mathematical Society 2003

  16. Cannon, J., Thurston, W.: Group invariant Peano curves. Preprint circa 1985

  17. Casson, A., Bleiler, S.: Automorphisms of surfaces after Nielsen and Thurston. LMS student texts, vol. 9. Cambridge: Cambridge University Press 1988

  18. Fenley, S.: Anosov flows in 3-manifolds. Ann. Math. 139, 79–115 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  19. Fenley, S.: The structure of branching in Anosov flows of 3-manifolds. Comment. Math. Helv. 73, 259–297 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  20. Fenley, S.: Foliations with good geometry. J. Am. Math. Soc. 12, 619–676 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  21. Gabai, D.: Foliations and the topology of 3-manifolds. J. Differ. Geom. 18, 445–503 (1983)

    MATH  MathSciNet  Google Scholar 

  22. Gabai, D.: Problems in foliations and laminations. Geometric Topology (Athens, GA, 1993) AMS/IP Stud. Adv. math. 2.2 1–33

  23. Gabai, D.: Essential laminations and Kneser normal form. J. Differ. Geom. 53, 517–574 (1999)

    MATH  MathSciNet  Google Scholar 

  24. Gabai, D., Kazez, W.: Order trees and laminations of the plane. Math. Res. Lett. 4, 603–616 (1997)

    MATH  MathSciNet  Google Scholar 

  25. Gabai, D., and Kazez, W.: Homotopy, isotopy and genuine laminations of 3-manifolds. In: Geometric topology (Athens, GA, 1993), pp. 123–138. Providence, RI: American Mathematical Society 1997

  26. Gabai, D., Kazez, W.: Group negative curvature for 3-manifolds with genuine laminations. Geom. Topol. 2, 65–77 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  27. Gabai, D., Kazez, W.: The finiteness of the mapping class group for atoroidal 3-manifolds with genuine laminations. J. Differ. Geom. 50, 123–127 (1998)

    MATH  MathSciNet  Google Scholar 

  28. Gabai, D., Oertel, U.: Essential laminations in 3-manifolds. Ann. Math. (2) 130, 41–73 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  29. Garnett, L.: Foliations, the ergodic theorem and Brownian motion. J. Funct. Anal. 51, 285–311 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  30. Gromov, M.: Hyperbolic groups. In: Essays in group theory, S. Gersten (ed.). MSRI, vol. 7. New York: Springer 1987

  31. Hass, J.: Minimal surfaces in foliated manifolds. Comment. Math. Helv. 61, 1–32 (1986)

    MATH  MathSciNet  Google Scholar 

  32. Hatcher, A.: Notes on basic 3-manifold topology. Cornell Mathematics Department webpage

  33. Hatcher, A., and Oertel, U.: Full laminations in 3-manifolds. Math. Proc. Camb. Philos. Soc. 119, 73–82 (1996)

    Article  MATH  Google Scholar 

  34. Hempel, J.: 3-Manifolds. Ann. Math. Stud., vol. 86. Princeton, NJ: Princeton University Press 1976

  35. Hocking, J., and Young, G.: Topology. Reading, MS: Addison-Wesley 1961

  36. Jaco, W.: Lectures on three-manifold topology. CBMS Regional Conference Series in Mathematics, vol. 43. Providence, RI: American Mathematical Society 1980

  37. Li, T.: Commutator subgroups and foliations without holonomy. Proc. Am. Math. Soc. 130, 2471–2477 (2002)

    Article  MATH  Google Scholar 

  38. Li, T.: Laminar branched surfaces in 3-manifolds. Geom. Topol. 6, 153–194 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  39. Lickorish, W.: A foliation for 3-manifolds. Ann. Math. (2) 82, 414–420 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  40. Moise, E.: Geometric Topology in Dimensions 2 and 3. Grad Texts Math., vol. 47. New York, Heidelberg: Springer 1977

  41. Mosher, L.: Laminations and flows transverse to finite depth foliations. Part I: Branched surfaces and dynamics. Preprint

  42. Novikov, S.: Topology of foliations. Trud. Mosc. Math. Ob. 14, 268–304 (1965)

    MATH  Google Scholar 

  43. Oertel, U.: Homology branched surfaces: Thurston’s norm on H2(M3). In: Low-dimensional topology and Kleinian Groups. LMS Ser. 112, 253–272 (1986)

    MATH  MathSciNet  Google Scholar 

  44. Palmeira, C.: Open manifolds foliated by planes. Ann. Math. 107, 109–131 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  45. Poincaré, H.: Sur les courbes définies par les équations diffèrentielles. J. Math. 1 (1885)

  46. Reeb, G.: Sur certaines proprietes topologiques des varietes feuilletees. Act. Sci. Indust. 1183, 91–158 (1952)

    MathSciNet  Google Scholar 

  47. Roberts, R., Shareshian, J., Stein, M.: Infinitely many hyperbolic 3-manifolds which contain no Reebless foliation. J. Am. Math. Soc. 16, 639–679 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  48. Rosenberg, H.: Foliations by planes. Topology 7, 131–138 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  49. Serre, J.P.: Trees. Berlin: Springer 1980

  50. Sierpinski, W.: Un theoreme sur les continus. Tohoku Math. J., II. Ser. 13, 300–303 (1918)

    MATH  Google Scholar 

  51. Sullivan, D.: Cycles for the dynamical study of foliated manifolds and complex manifolds. Invent. Math. 36, 225–255 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  52. Sullivan, D.: A homological characterization of foliations consisting of minimal surfaces. Comment. Math. Helv. 54, 218–223 (1979)

    MATH  MathSciNet  Google Scholar 

  53. Thurston, W.: A local construction of foliations for 3-manifolds. Proc. Symp. Pure Math. XXVII 1, 315–319 (1973)

    Google Scholar 

  54. Thurston, W.: A norm for the homology of 3-manifolds. Mem. Am. Math. Soc. 59, i–vi and 99–130 (1986)

  55. Thurston, W.: On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Am. Math. Soc. 19, 417–431 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  56. Thurston, W.: Three-Dimensional Geometry and Topology, vol. 1, S. Levy (ed.). Princeton Mathematical Series, vol. 35. Princeton, NJ: Princeton University Press 1997

  57. Thurston, W.: Notes by D. Calegari from talks at the very informal foliations seminar at MSRI, 1996–1997

  58. Thurston, W.: 3-manifolds, foliations and circles I. eprint math.GT/9712268

  59. Thurston, W.: 3-manifolds, foliations and circles II. Preprint

  60. Thurston, W.: Hyperbolic structures on 3-manifolds II: Surface groups and 3-manifolds which fiber over the circle. eprint math.GT/9801045

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Danny Calegari.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Calegari, D. Promoting essential laminations. Invent. math. 166, 583–643 (2006). https://doi.org/10.1007/s00222-006-0004-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-006-0004-3

Keywords

Navigation