On Ahlfors’ finiteness theorem.

*(English)*Zbl 0684.57019Let \(\Gamma\) 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.

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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\textit{R. S. Kulkarni} and \textit{P. B. Shalen}, Adv. Math. 76, No. 2, 155--169 (1989; Zbl 0684.57019)

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##### References:

[1] | Ahlfors, L, Finitely generated Kleinian groups, Amer. J. math., 86, 413-429, (1964) · Zbl 0133.04201 |

[2] | Bers, L, On Ahlfors’ finiteness theorem, Amer. J. math., 89, 1078-1082, (1967) · Zbl 0167.07003 |

[3] | Bers, L, Inequalities for finitely generated Kleinian groups, J. analyse math., 18, 23-41, (1967) · Zbl 0146.31601 |

[4] | Freudenthal, H, Über die enden topologischer Räume und gruppen, Math. Z., 33, 692-713, (1931) · JFM 57.0731.01 |

[5] | Greenberg, L, On a theorem of Ahlfors and conjugate subgroups of Kleinian groups, Amer. J. math., 89, 56-68, (1967) · Zbl 0154.33605 |

[6] | Hempel, J, 3-manifolds, () · Zbl 0191.22203 |

[7] | Hopf, H, Enden offener Räume und unendliche discontinuierliche gruppen, Comment. math. helv., 16, 81-100, (1943-1944) · Zbl 0060.40008 |

[8] | Jaco, W, Finitely presented subgroups of 3-manifold groups, Invent. math., 13, 335-346, (1971) · Zbl 0232.57003 |

[9] | Jaco, W, Lectures on three-manifold topology, () · Zbl 0433.57001 |

[10] | Kulkarni, R.S, Some topological aspects of Kleinian groups, Amer. J. math., 100, 897-911, (1978) · Zbl 0455.57022 |

[11] | Kulkarni, R.S, Groups with domains of discontinuity, Math. ann., 237, 253-272, (1978) · Zbl 0369.20028 |

[12] | Marden, A, The geometry of finitely generated Kleinian groups, Ann. of math., 99, 383-462, (1974) · Zbl 0282.30014 |

[13] | Scott, G.P, Finitely generated 3-manifold groups are finitely presented, J. London math. soc., 6, 2, 437-440, (1973) · Zbl 0254.57003 |

[14] | Sullivan, D, A finiteness theorem for cusps, Acta math., 147, 289-299, (1981) · Zbl 0502.57004 |

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