Skip to main content
Log in

Heegaard Splittings and Pseudo-Anosov Maps

  • Published:
Geometric and Functional Analysis Aims and scope Submit manuscript

Abstract

Given two 3-dimensional handlebodies whose boundaries are identified with a surface S of genus g > 1 and with different orientations, we consider the sequence of manifolds M n obtained by gluing the handlebodies via the iteration f n of a “generic” pseudo-Anosov homeomorphism f of S. Using the deformation theory of hyperbolic structures on open hyperbolic 3-manifolds and for n sufficiently large, we construct a negatively curved metric on M n where the sectional curvatures are pinched in a given small interval centered at –1. The construction is concrete enough to allow us describe the geometric limits of these manifolds as n tends to infinity and the metrics get closer to being hyperbolic. Such a description allows us to prove various topological and group theoretical properties of M n , for n sufficiently large, which would not be available knowing the mere existence of a negatively curved or even hyperbolic metric on M n .

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. I. Agol, Tameness of hyperbolic 3-manifolds, preprint (2004).

  2. R. Benedetti, C. Petronio, Lectures on Hyperbolic Geometry, Springer-Verlag, 1992.

  3. Bestvina M.: Degenerations of the hyperbolic space. Duke Math. J. 56, 143–161 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bonahon F.: Cobordism of automorphisms of surfaces, Ann. Sci. Ec. Norm. Super. IV Ser. 16, 237–270 (1983)

    MATH  MathSciNet  Google Scholar 

  5. Bonahon F.: Bouts des variété hyperboliques de dimension 3. Ann. of Math. 124, 71–158 (1986)

    Article  MathSciNet  Google Scholar 

  6. Brock J.: Continuity of Thurston’s length function. Geom. Funct. Anal. 10, 741–797 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  7. D. Calegari, D. Gabai, Shrinkwrapping and the taming of hyperbolic 3-manifolds, preprint (2004).

  8. Canary R.D.: Ends of hyperbolic 3-manifolds. J. Amer. Math. Soc. 6, 1–35 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  9. Canary R.D.: A covering theorem for hyperbolic 3-manifolds and its applications. Topology 35, 751–778 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  10. R.D. Canary, D.B.A. Epstein, P. Green, Notes on notes of Thurston, in “Analytical and geometric aspects of hyperbolic space”, London Math. Soc. Lecture Note Ser. 111, Cambridge University Press (1987), 3–92.

  11. Casson A., Gordon C.: Reducing Heegaard splittings. Topology Appl. 27, 275–283 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  12. Choi H.I., Schoen R.: The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature. Invent. Math. 81, 387–394 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  13. Easson V.R.: Surface subgroups and handlebody attachments. Geom. Topol. 10, 557–591 (2006) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hartshorn K.: Heegaard splittings of Haken manifolds have bounded distance. Pacific J. Math. 204, 61–75 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. J. Hass, A. Thompson, W. Thurston, Stabilization of Heegaard splittings, preprint; arXiv: 0802.2145.

  16. Hempel J.: 3-manifolds as viewed from the curve complex. Topology 40, 631–657 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kerckhoff S.: The measure of the limit set of the handlebody group. Topology 29, 27–40 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kleineidam G., Souto J.: Algebraic convergence of function groups. Comment. Math. Helv. 77, 244–269 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. G. Kleineidam, J. Souto, Ending laminations in theMasur domain, in “Kleinian Groups and Hyperbolic 3-Manifolds,” Poceeddings of Warwick Conference 2001, London Math. Soc. (2003), 105–129.

  20. Lackenby M.: Attaching handlebodies to 3-manifolds. Geom. Topol. 6, 889–904 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  21. Lackenby M.: Heegaard splittings, the virtually Haken conjecture and Property τ. Invent. Math. 164, 317–359 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  22. Masur H.A.: Measured foliations and handlebodies. Ergodic Theory Dynam. Systems 6, 99–116 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  23. K. Matsuzaki, M. Taniguchi, Hyperbolic Manifolds and Kleinian Groups, Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, (1998).

  24. McMullen C.: Renormalization and 3-Manifolds Which Fiber over the Circle. Princeton University Press, Princeton, NJ (1996)

    MATH  Google Scholar 

  25. Minsky Y.N.: Bounded geometry for Kleinian groups. Invent. Math. 146, 143–192 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  26. H. Namazi, Heegaard splittings and hyperbolic geometry, Dissertation Research, Stony Brook University, 2005.

  27. J.-P. Otal, Courants géodésiques et produits libres, Thése d’Etat, Université Paris-Sud, Orsay, 1988.

  28. Otal J.-P.: Sur la dégénerescence des groupes de Schottky. Duke Math. J. 74, 777–792 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  29. J.-P. Otal, Théor‘eme d’hyperbolisation pour les variété fibrées de dimension 3, Astérisque, Société Mathématique de France, (1996).

  30. Paulin F.: Topologie de Gromov équivariante, structures hyperboliques et arbres réels. Invent. Math. 94, 53–80 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  31. J. Pérez, A. Ros, Properly embedded minimal surfaces with finite total curvature, in “The global theory of minimal surfaces in flat spaces”, Springer Lecture Notes in Math. 1775 (2002), 15–66.

  32. Sacks J., Uhlenbeck K.: Minimal immersions of closed riemann surfaces. Trans. Amer. Math. Soc. 271, 639–652 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  33. Scharlemann M., Tomova M.: Alternate Heegaard genus bounds distance. Geom. Topol. 10, 593–617 (2006) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  34. Schoen R., Yau S.T.: Existence of incompressible minimal surfaces and the topology of three dimensional manifolds with non-negative scalar curvature. Annals of Math. 110, 127–142 (1979)

    Article  MathSciNet  Google Scholar 

  35. P. Scott, Compact submanifolds of 3-manifolds, Journal London Math. Soc. 7 (1973), .

  36. Skora R.: Splittings of surfaces. Bull. Amer. Math. Soc. 23, 85–90 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  37. Skora R.: Splittings of surfaces. J. Amer. Math. Soc. 9, 605–616 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  38. J. Souto, The rank of the fundamental group of hyperbolic 3-manifolds fibering over the circle, in “The Zieschang Gedenkschrift”, Geometry and Topology Monographs 14 (2008).

  39. J. Souto, Geometry of Heegaard splittings, in preparation.

  40. Thurston W.P.: Hyperbolic structures on 3-manifolds I: Deformation of acylindrical manifolds. Annals of Math. 124, 203–246 (1986)

    Article  MathSciNet  Google Scholar 

  41. W.P. Thurston, Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over he circle, preprint; math.GT/9801045

  42. G. Tian, A pinching theorem on manifolds with negative curvature, Proceedings of International Conference on Algebraic and Analytic Geometry, Tokyo (1990), .

  43. Waldhausen F.: Heegaard–Zerlegungen der 3-Sphäre. Topology 7, 195–203 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  44. White M.: Injectivity radius and fundamental groups of hyperbolic 3-manifolds. Comm. Anal. Geom. 10, 377–395 (2002)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hossein Namazi.

Additional information

The first author was partially supported by an award of the Clay Mathematics Institute and by National Science Foundation Grant 0604111. The second author was partially supported by National Science Foundation Grant 0706878 and Alfred P. Sloan Foundation.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Namazi, H., Souto, J. Heegaard Splittings and Pseudo-Anosov Maps. Geom. Funct. Anal. 19, 1195–1228 (2009). https://doi.org/10.1007/s00039-009-0025-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00039-009-0025-3

Keywords and phrases

2000 Mathematics Subject Classification

Navigation