Tameness on the boundary and Ahlfors' measure conjecture

Details Tameness on the boundary and Ahlfors' measure conjecture Tameness on the boundary and Ahlfors' measure conjecture

2002-11-01

arxiv.org/abs/math/0211022v2

Mathematics

Geometric Topology

New revised version, 22 pages. To appear, Publ. I.H.E.S. This version represents a fairly substantial reorganization of the lo

Scientific paper

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 G of geometrically finite Kleinian groups, the limit set of G is either of Lebesgue measure zero or all of the Riemann sphere. Thus, Ahlfors' conjecture is reduced to the density conjecture of Bers, Sullivan, and Thurston.

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