Tameness on the boundary and Ahlfors’ measure conjecture

  • Jeffrey Brock
  • Kenneth Bromberg
  • Richard Evans
  • Juan Souto
Article

Abstract

Let N be a complete hyperbolic 3-manifold that is an algebraic limit of geometrically finite hyperbolic 3-manifolds. We show N is homeomorphic to the interior of a compact 3-manifold, or tame, if one of the following conditions holds:

1. N has non-empty conformal boundary,

2. N is not homotopy equivalent to a compression body, or

3. N is a strong limit of geometrically finite manifolds.

The first case proves Ahlfors’ measure conjecture for Kleinian groups in the closure of the geometrically finite locus: given any algebraic limit Γ of geometrically finite Kleinian groups, the limit set of Γ is either of Lebesgue measure zero or all of Ĉ. Thus, Ahlfors’ conjecture is reduced to the density conjecture of Bers, Sullivan, and Thurston.

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References

  1. 1.
    W. Abikoff, Degenerating families of Riemann surfaces, Ann. Math., 105 (1977), 29–44. Google Scholar
  2. 2.
    L. Ahlfors, Finitely generated Kleinian groups, Am. J. Math., 86 (1964), 413–429. Google Scholar
  3. 3.
    L. Ahlfors, Fundamental polyhedrons and limit point sets of Kleinian groups, Proc. Natl. Acad. Sci. USA, 55 (1966), 251–254. Google Scholar
  4. 4.
    J. Anderson and R. Canary, Cores of hyperbolic 3-manifolds and limits of Kleinian groups, Am. J. Math., 118 (1996), 745–779. Google Scholar
  5. 5.
    J. Anderson and R. Canary, Cores of hyperbolic 3-manifolds and limits of Kleinian groups II, J. Lond. Math. Soc., 61 (2000), 489–505. Google Scholar
  6. 6.
    R. Benedetti and C. Petronio, Lectures on Hyperbolic Geometry, Springer-Verlag, 1992. Google Scholar
  7. 7.
    F. Bonahon, Cobordism of automorphisms of surfaces, Ann. Sci. Éc. Norm. Supér., 16 (1983), 237–270. Google Scholar
  8. 8.
    F. Bonahon, Bouts des variétés hyperboliques de dimension 3, Ann. Math., 124 (1986), 71–158. Google Scholar
  9. 9.
    F. Bonahon and J. P. Otal, Variétés hyperboliques à géodésiques arbitrairement courtes, Bull. Lond. Math. Soc., 20 (1988), 255–261. Google Scholar
  10. 10.
    J. Brock, Iteration of mapping classes and limits of hyperbolic 3-manifolds, Invent. Math., 143 (2001), 523–570. Google Scholar
  11. 11.
    J. Brock and K. Bromberg, Cone Manifolds and the Density Conjecture, To appear in the proceedings of the Warwick conference ‘Kleinian groups and hyperbolic 3-manifolds,’ arXiv:mathGT/0210484 (2002). Google Scholar
  12. 12.
    J. Brock and K. Bromberg, On the density of geometrically finite Kleinian groups, Accepted to Acta Math., arXiv:mathGT/0212189 (2002). Google Scholar
  13. 13.
    K. Bromberg, Hyperbolic Dehn surgery on geometrically infinite 3-manifolds, Preprint (2000). Google Scholar
  14. 14.
    K. Bromberg, Rigidity of geometrically finite hyperbolic cone-manifolds, To appear, Geom. Dedicata, arXiv:mathGT/0009149 (2000). Google Scholar
  15. 15.
    K. Bromberg, Hyperbolic cone manifolds, short geodesics, and Schwarzian derivatives, Preprint, arXiv:mathGT/0211401 (2002). Google Scholar
  16. 16.
    K. Bromberg, Projective structures with degenerate holonomy and the Bers density conjecture, Preprint, arXiv:mathGT/0211402 (2002). Google Scholar
  17. 17.
    R. Brooks and J. P. Matelski, Collars for Kleinian Groups, Duke Math. J., 49 (1982), 163–182. Google Scholar
  18. 18.
    R. D. Canary, Ends of hyperbolic 3-manifolds, J. Am. Math. Soc., 6 (1993), 1–35. Google Scholar
  19. 19.
    R. D. Canary, Geometrically tame hyperbolic 3-manifolds, In Differential geometry: Riemannian geometry (Los Angeles, CA, 1990), volume 54 of Proc. Symp. Pure Math., pp. 99–109. Am. Math. Soc., 1993. Google Scholar
  20. 20.
    R. D. Canary, D. B. A. Epstein and P. Green, Notes on notes of Thurston. In Analytical and Geometric Aspects of Hyperbolic Space, pp. 3–92. Cambridge University Press, 1987. Google Scholar
  21. 21.
    R. D. Canary and Y. N. Minsky, On limits of tame hyperbolic 3-manifolds, J. Differ. Geom., 43 (1996), 1–41. Google Scholar
  22. 22.
    C. J. Earle and A. Marden, Geometric complex coordinates for Teichmüller space, In preparation. Google Scholar
  23. 23.
    R. Evans, Deformation spaces of hyperbolic 3-manifolds: strong convergence and tameness, Ph.D. Thesis, Unversity of Michigan (2000). Google Scholar
  24. 24.
    R. Evans. Tameness persists in weakly type-preserving strong limits, Preprint. Google Scholar
  25. 25.
    C. Hodgson and S. Kerckhoff, Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery, J. Differ. Geom., 48 (1998), 1–59. Google Scholar
  26. 26.
    C. Hodgson and S. Kerckhoff, Universal bounds for hyperbolic Dehn surgery, Preprint, arXiv:math.GT/0204345 (2002). Google Scholar
  27. 27.
    C. Hodgson and S. Kerckhoff, The shape of hyperbolic Dehn surgery space, In preparation (2002). Google Scholar
  28. 28.
    S. Kerckhoff and W. Thurston, Non-continuity of the action of the modular group at Bers’ boundary of Teichmüller space, Invent. Math., 100 (1990), 25–48. Google Scholar
  29. 29.
    R. Kulkarni and P. Shalen, On Ahlfors’ finiteness theorem, Adv. Math., 76 (1989), 155–169. Google Scholar
  30. 30.
    A. Marden, The geometry of finitely generated kleinian groups, Ann. Math., 99 (1974), 383–462. Google Scholar
  31. 31.
    A. Marden, Geometrically finite Kleinian groups and their deformation spaces, In Discrete groups and automorphic functions, Academic Press (1977), pp. 259–293. Academic Press, 1977. Google Scholar
  32. 32.
    B. Maskit, On boundaries of Teichmüller spaces and on kleinian groups: II, Ann. Math., 91 (1970), 607–639. Google Scholar
  33. 33.
    B. Maskit, Kleinian Groups, Springer-Verlag, 1988. Google Scholar
  34. 34.
    D. McCullough, Compact submanifolds of 3-manifolds with boundary, Quart. J. Math. Oxf., 37 (1986), 299–307. Google Scholar
  35. 35.
    C. McMullen, The classification of conformal dynamical systems, In Current Developments in Mathematics, 1995, pp. 323–360. International Press, 1995. Google Scholar
  36. 36.
    C. McMullen, Renormalization and 3-Manifolds Which Fiber Over the Circle, Annals of Math Studies 142, Princeton University Press, 1996. Google Scholar
  37. 37.
    D. Mostow, Strong rigidity of locally symmetric spaces, Annals of Math. Studies 78, Princeton University Press, 1972. Google Scholar
  38. 38.
    K. Ohshika, Kleinian groups which are limits of geometrically finite groups, Preprint. Google Scholar
  39. 39.
    P. Scott, Compact submanifolds of 3-manifolds, J. Lond. Math. Soc., (2) 7 (1973), 246–250. Google Scholar
  40. 40.
    D. Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, In Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference. Annals of Math. Studies 97, Princeton, 1981. Google Scholar
  41. 41.
    D. Sullivan, Quasiconformal homeomorphisms and dynamics II: Structural stability implies hyperbolicity for Kleinian groups, Acta Math., 155 (1985), 243–260. Google Scholar
  42. 42.
    W. P. Thurston, Geometry and Topology of Three-Manifolds, Princeton Lecture Notes, 1979. Google Scholar
  43. 43.
    W. P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Am. Math. Soc., 6 (1982), 357–381. Google Scholar
  44. 44.
    W. P. Thurston, Hyperbolic structures on 3-manifolds I: Deformations of acylindrical manifolds, Ann. Math., 124 (1986), 203–246. Google Scholar
  45. 45.
    W. P. Thurston, Hyperbolic structures on 3-manifolds II: Surface groups and 3-manifolds which fiber over the circle, Preprint, arXiv:math.GT/9801045 (1986).Google Scholar

Copyright information

© Institut des Hautes Études Scientifiques and Springer-Verlag 2003

Authors and Affiliations

  • Jeffrey Brock
    • 1
  • Kenneth Bromberg
    • 2
  • Richard Evans
    • 3
  • Juan Souto
    • 4
  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA
  2. 2.Department of MathematicsCalifornia Institute of TechnologyPasadenaUSA
  3. 3.Department of MathematicsRice UniversityHoustonUSA
  4. 4.Mathematisches InstitutUniverstät BonnBonnGermany

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