Tameness on the boundary and Ahlfors’ measure conjecture

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

DOI: 10.1007/s10240-003-0018-y

Cite this article as:
Brock, J., Bromberg, K., Evans, R. et al. Publ. Math. (2003) 98: 145. doi:10.1007/s10240-003-0018-y

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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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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