Tameness on the boundary and Ahlfors' measure conjecture
Publications Mathématiques de l'IHÉS, Volume 98  (2003), p. 145-166

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 𝐂 ^. Thus, Ahlfors’ conjecture is reduced to the density conjecture of Bers, Sullivan, and Thurston.

@article{PMIHES_2003__98__145_0,
     author = {Brock, Jeffrey and Bromberg, Kenneth and Evans, Richard and Souto, Juan},
     title = {Tameness on the boundary and Ahlfors' measure conjecture},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     publisher = {Springer},
     volume = {98},
     year = {2003},
     pages = {145-166},
     doi = {10.1007/s10240-003-0018-y},
     zbl = {1060.30054},
     language = {en},
     url = {http://www.numdam.org/item/PMIHES_2003__98__145_0}
}
Brock, Jeffrey; Bromberg, Kenneth; Evans, Richard; Souto, Juan. Tameness on the boundary and Ahlfors' measure conjecture. Publications Mathématiques de l'IHÉS, Volume 98 (2003) , pp. 145-166. doi : 10.1007/s10240-003-0018-y. http://www.numdam.org/item/PMIHES_2003__98__145_0/

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