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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Brock, J., Bromberg, K., Evans, R. et al. Tameness on the boundary and Ahlfors’ measure conjecture. Publ. Math. 98, 145–166 (2003). https://doi.org/10.1007/s10240-003-0018-y
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DOI: https://doi.org/10.1007/s10240-003-0018-y