Heegaard Splittings and Pseudo-Anosov Maps

Article

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 Mn obtained by gluing the handlebodies via the iteration fn 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 Mn 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 Mn, for n sufficiently large, which would not be available knowing the mere existence of a negatively curved or even hyperbolic metric on Mn.

Keywords and phrases

Heegaard splitting geometrization hyperbolic 3-manifolds rank conjecture 

2000 Mathematics Subject Classification

57M50 57M07 30F40 57N10 

References

  1. A.
    I. Agol, Tameness of hyperbolic 3-manifolds, preprint (2004).Google Scholar
  2. BP.
    R. Benedetti, C. Petronio, Lectures on Hyperbolic Geometry, Springer-Verlag, 1992.Google Scholar
  3. Be.
    Bestvina M.: Degenerations of the hyperbolic space. Duke Math. J. 56, 143–161 (1988)MATHCrossRefMathSciNetGoogle Scholar
  4. Bo1.
    Bonahon F.: Cobordism of automorphisms of surfaces, Ann. Sci. Ec. Norm. Super. IV Ser. 16, 237–270 (1983)MATHMathSciNetGoogle Scholar
  5. Bo2.
    Bonahon F.: Bouts des variété hyperboliques de dimension 3. Ann. of Math. 124, 71–158 (1986)CrossRefMathSciNetGoogle Scholar
  6. Br.
    Brock J.: Continuity of Thurston’s length function. Geom. Funct. Anal. 10, 741–797 (2000)MATHCrossRefMathSciNetGoogle Scholar
  7. CG.
    D. Calegari, D. Gabai, Shrinkwrapping and the taming of hyperbolic 3-manifolds, preprint (2004).Google Scholar
  8. Ca1.
    Canary R.D.: Ends of hyperbolic 3-manifolds. J. Amer. Math. Soc. 6, 1–35 (1993)MATHCrossRefMathSciNetGoogle Scholar
  9. Ca2.
    Canary R.D.: A covering theorem for hyperbolic 3-manifolds and its applications. Topology 35, 751–778 (1996)MATHCrossRefMathSciNetGoogle Scholar
  10. CaEG.
    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.Google Scholar
  11. CasG.
    Casson A., Gordon C.: Reducing Heegaard splittings. Topology Appl. 27, 275–283 (1987)MATHCrossRefMathSciNetGoogle Scholar
  12. ChS.
    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)MATHCrossRefMathSciNetGoogle Scholar
  13. E.
    Easson V.R.: Surface subgroups and handlebody attachments. Geom. Topol. 10, 557–591 (2006) (electronic)MATHCrossRefMathSciNetGoogle Scholar
  14. H.
    Hartshorn K.: Heegaard splittings of Haken manifolds have bounded distance. Pacific J. Math. 204, 61–75 (2002)MATHCrossRefMathSciNetGoogle Scholar
  15. HaTT.
    J. Hass, A. Thompson, W. Thurston, Stabilization of Heegaard splittings, preprint; arXiv: 0802.2145.Google Scholar
  16. He.
    Hempel J.: 3-manifolds as viewed from the curve complex. Topology 40, 631–657 (2001)MATHCrossRefMathSciNetGoogle Scholar
  17. K.
    Kerckhoff S.: The measure of the limit set of the handlebody group. Topology 29, 27–40 (1990)MATHCrossRefMathSciNetGoogle Scholar
  18. KlS1.
    Kleineidam G., Souto J.: Algebraic convergence of function groups. Comment. Math. Helv. 77, 244–269 (2002)MATHCrossRefMathSciNetGoogle Scholar
  19. KlS2.
    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.Google Scholar
  20. L1.
    Lackenby M.: Attaching handlebodies to 3-manifolds. Geom. Topol. 6, 889–904 (2002)MATHCrossRefMathSciNetGoogle Scholar
  21. L2.
    Lackenby M.: Heegaard splittings, the virtually Haken conjecture and Property τ. Invent. Math. 164, 317–359 (2006)MATHCrossRefMathSciNetGoogle Scholar
  22. M.
    Masur H.A.: Measured foliations and handlebodies. Ergodic Theory Dynam. Systems 6, 99–116 (1986)MATHCrossRefMathSciNetGoogle Scholar
  23. MaT98.
    K. Matsuzaki, M. Taniguchi, Hyperbolic Manifolds and Kleinian Groups, Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, (1998).Google Scholar
  24. Mc.
    McMullen C.: Renormalization and 3-Manifolds Which Fiber over the Circle. Princeton University Press, Princeton, NJ (1996)MATHGoogle Scholar
  25. Mi.
    Minsky Y.N.: Bounded geometry for Kleinian groups. Invent. Math. 146, 143–192 (2001)MATHCrossRefMathSciNetGoogle Scholar
  26. N.
    H. Namazi, Heegaard splittings and hyperbolic geometry, Dissertation Research, Stony Brook University, 2005.Google Scholar
  27. O1.
    J.-P. Otal, Courants géodésiques et produits libres, Thése d’Etat, Université Paris-Sud, Orsay, 1988.Google Scholar
  28. O2.
    Otal J.-P.: Sur la dégénerescence des groupes de Schottky. Duke Math. J. 74, 777–792 (1994)MATHCrossRefMathSciNetGoogle Scholar
  29. O3.
    J.-P. Otal, Théor‘eme d’hyperbolisation pour les variété fibrées de dimension 3, Astérisque, Société Mathématique de France, (1996).Google Scholar
  30. P.
    Paulin F.: Topologie de Gromov équivariante, structures hyperboliques et arbres réels. Invent. Math. 94, 53–80 (1988)MATHCrossRefMathSciNetGoogle Scholar
  31. PeR.
    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.Google Scholar
  32. SU.
    Sacks J., Uhlenbeck K.: Minimal immersions of closed riemann surfaces. Trans. Amer. Math. Soc. 271, 639–652 (1982)MATHCrossRefMathSciNetGoogle Scholar
  33. ScT.
    Scharlemann M., Tomova M.: Alternate Heegaard genus bounds distance. Geom. Topol. 10, 593–617 (2006) (electronic)MATHCrossRefMathSciNetGoogle Scholar
  34. SchY.
    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)CrossRefMathSciNetGoogle Scholar
  35. Sco.
    P. Scott, Compact submanifolds of 3-manifolds, Journal London Math. Soc. 7 (1973), .Google Scholar
  36. Sk1.
    Skora R.: Splittings of surfaces. Bull. Amer. Math. Soc. 23, 85–90 (1990)MATHCrossRefMathSciNetGoogle Scholar
  37. Sk2.
    Skora R.: Splittings of surfaces. J. Amer. Math. Soc. 9, 605–616 (1996)MATHCrossRefMathSciNetGoogle Scholar
  38. So1.
    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).Google Scholar
  39. So2.
    J. Souto, Geometry of Heegaard splittings, in preparation.Google Scholar
  40. T1.
    Thurston W.P.: Hyperbolic structures on 3-manifolds I: Deformation of acylindrical manifolds. Annals of Math. 124, 203–246 (1986)CrossRefMathSciNetGoogle Scholar
  41. T2.
    W.P. Thurston, Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over he circle, preprint; math.GT/9801045Google Scholar
  42. Ti.
    G. Tian, A pinching theorem on manifolds with negative curvature, Proceedings of International Conference on Algebraic and Analytic Geometry, Tokyo (1990), .Google Scholar
  43. W.
    Waldhausen F.: Heegaard–Zerlegungen der 3-Sphäre. Topology 7, 195–203 (1968)MATHCrossRefMathSciNetGoogle Scholar
  44. Wh.
    White M.: Injectivity radius and fundamental groups of hyperbolic 3-manifolds. Comm. Anal. Geom. 10, 377–395 (2002)MATHMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TexasAustinUSA
  2. 2.Department of MathematicsUniversity of MichiganAnn ArborUSA

Personalised recommendations