Publications Mathématiques de l'Institut des Hautes Études Scientifiques

, Volume 98, Issue 1, pp 145–166

Tameness on the boundary and Ahlfors’ measure conjecture

Authors

    • Department of MathematicsUniversity of Chicago
  • Kenneth Bromberg
    • Department of MathematicsCalifornia Institute of Technology
  • Richard Evans
    • Department of MathematicsRice University
  • Juan Souto
    • Mathematisches InstitutUniverstät Bonn
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.

Copyright information

© Institut des Hautes Études Scientifiques and Springer-Verlag 2003