On Ahlfors' finiteness theorem

On the Ahlfors Finiteness Theorem

The goal of this note is to give a proof of the Ahlfors Finiteness theorem which requires just the bare minimum of the complex analysis: (a) the existence theorem for the Beltrami equation and (b) the Rado-Cartan uniqueness theorem for holomorphic functions. However our proof does require some (by now standard) 3-dimensional topology and Greenberg's algebraic trick to deal with the triply-punct...

Generation of the Ahlfors Five Islands Theorem

A Theorem of Ahlfors for Hyperbolic Spaces

L. Ahlfors has proved that if the Dirichlet fundamental polyhedron of a Kleinian group G in the unit ball B3 has finitely many sides, then the normalized Lebesgue measure of L(G) is either zero or one. We generalize this theorem and a theorem of Beardon and Maskit to the n-dimensional case.

generation of the ahlfors five islands theorem

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NEW PROOF OF THE AHLFORS FIVE ISLANDS THEOREM 339 from

Let D j, j = 1 , . . . , 5, be simply-connected domains on the Riemann sphere with piecewise analytic boundary and pairwise disjoint closures. Let D C C be a domain and denote by 3rA(D) = f a (D, {Dj}~=I) the family of all meromorphic functions f : D ~ C with the property that no subdomain of D is mapped conformally onto one of the domains Dj by f . (If there is such a subdomain, then it is cal...