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On Ahlfors’ finiteness theorem. (English) Zbl 0684.57019
Let $$\Gamma$$ denote a Kleinian group. Let $$\Lambda$$ denote the limit set of $$\Gamma$$ (the closure of the set of fixed points of the elements of $$\Gamma$$ of infinite order). Let $$\Omega$$ denote $$S^ 2-\Lambda$$. The theorem of the title is the following: If $$\Gamma$$ is finitely generated and $$\Lambda$$ has more than two points, then $$\Gamma$$ $$\setminus \Omega$$ has finitely many components, each of which, as an orbifold, is a hyperbolic Riemann surface of finite type. Supplements to this theorem were proved by Bers and Sullivan who have given theorems which give more explicit information about $$\Gamma$$ and its action.
The aim of the authors is to provide a (3-dimensional) topological context to better understand these theorems. Their main theorem and proof contain explicit topological conclusions which provide alternate proofs for (and stronger results than) the earlier cited results.
Reviewer: L.Neuwirth

MSC:
 57S30 Discontinuous groups of transformations 57N05 Topology of the Euclidean $$2$$-space, $$2$$-manifolds (MSC2010) 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
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References:
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