Kleinian Groups Which are Limits of Geometrically Finite Groups

Ahlfors conjectured in 1964 that the limit set of every finitely generated Kleinian group either has Lebesgue measure $0$ or is the entire $S^2$. This title intends to prove that this conjecture is true for purely loxodromic Kleinian groups which are algebraic limits of geometrically finite groups.

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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 ${/mathbf H}^3[LAMBDA]Gamma$. The proof uses techniques of hyperbolic geometry considerably and is based on works of Maskit, Thurston, Bonahon, Otal, and Canary.