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Kleinian Groups which Are Limits of Geometrically Finite Groups
Ken'ichi Ohshika, Osaka University, Japan
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Memoirs of the American Mathematical Society
2005; 116 pp; softcover
Volume: 177
ISBN-10: 0-8218-3772-9
ISBN-13: 978-0-8218-3772-6
List Price: US$64 Individual Members: US$38.40
Institutional Members: US\$51.20
Order Code: MEMO/177/834

Ahlfors conjectured in 1964 that the limit set of every finitely generated Kleinian group either has Lebesgue measure $$0$$ or is the entire $$S^2$$. We prove that this conjecture is true for purely loxodromic Kleinian groups which are algebraic limits of geometrically finite groups. What we directly prove is that if a purely loxodromic Kleinian group $$\Gamma$$ is an algebraic limit of geometrically finite groups and the limit set $$\Lambda_\Gamma$$ is not the entire $$S^2_\infty$$, then $$\Gamma$$ is topologically (and geometrically) tame, that is, there is a compact 3-manifold whose interior is homeomorphic to $${\mathbf H}^3/\Gamma$$. The proof uses techniques of hyperbolic geometry considerably and is based on works of Maskit, Thurston, Bonahon, Otal, and Canary.