an:02111679
Zbl 1060.30054
Brock, Jeffrey; Bromberg, Kenneth; Evans, Richard; Souto, Juan
Tameness on the boundary and Ahlfors' measure conjecture
EN
Publ. Math., Inst. Hautes Étud. Sci. 98, 145-166 (2003).
0073-8301 1618-1913
2003
j
30F40 57M50 57N10
Marden's tameness conjecture; Ahlfors measure conjecture; hyperbolic 3-manifold
A complete hyperbolic 3-manifold is said to be tame if it is homeomorphic to the interior of a compact 3-manifold. Marden's tameness conjecture is that a complete hyperbolic 3-manifold with finitely generated fundamental group is tame. In this paper the authors show that a complete hyperbolic 3-manifold \(N\) which is an algebraic limit of geometrically finite hyperbolic 3-manifolds is tame if \(N\) has non-empty conformal boundary. This result reduces the Ahlfors' measure conjecture to the following density conjecture: a complete hyperbolic 3-manifold with finitely generated fundamental group is an algebraic limit of geometrically finite hyperbolic 3-manifolds. The key theorem is that an algebraic limit of geometrically finite hyperbolic 3-manifolds is a limit of a type-preserving sequence of geometrically finite hyperbolic 3-manifolds. The authors also show the tameness with respect to a compression body and a strong limit of geometrically finite manifolds.
Gou Nakamura (Toyota)