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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

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)
Full Text: DOI
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