Topologically conjugate Kleinian groups.

*(English)*Zbl 0845.30032Let \(\Gamma_1\) and \(\Gamma_2\) be finitely generated, torsion free Kleinian groups (acting on the 3-dimensional hyperbolic space and on its boundary \(S^2\)). We suppose that both \(\Gamma_1\) and \(\Gamma_2\) have positive injectivity radius. We also suppose that neither \(\Gamma_1\) and \(\Gamma_2\) can be expressed as a free product of two of their subgroups. Then the author poves that if there is a homeomorphism conjugating the action of \(\Gamma_1\) on \(S^2\) to that of \(\Gamma_2\) there is a quasi-conformal homeomorphism with the same property. The theorem is deduced from a theorem of Minsky by techniques of Bonahon and Waldhausen.

Reviewer: S.J.Patterson (Göttingen)

##### MSC:

30F40 | Kleinian groups (aspects of compact Riemann surfaces and uniformization) |

57M50 | General geometric structures on low-dimensional manifolds |

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\textit{K. Ohshika}, Proc. Am. Math. Soc. 124, No. 3, 739--743 (1996; Zbl 0845.30032)

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##### References:

[1] | Francis Bonahon, Bouts des variétés hyperboliques de dimension 3, Ann. of Math. (2) 124 (1986), no. 1, 71 – 158 (French). · Zbl 0671.57008 |

[2] | John Hempel, 3-Manifolds, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. · Zbl 0345.57001 |

[3] | Albert Marden, The geometry of finitely generated kleinian groups, Ann. of Math. (2) 99 (1974), 383 – 462. · Zbl 0282.30014 |

[4] | Yair N. Minsky, On rigidity, limit sets, and end invariants of hyperbolic 3-manifolds, J. Amer. Math. Soc. 7 (1994), no. 3, 539 – 588. · Zbl 0808.30027 |

[5] | Ken’ichi Ohshika, Geometric behaviour of Kleinian groups on boundaries for deformation spaces, Quart. J. Math. Oxford Ser. (2) 43 (1992), no. 169, 97 – 111. · Zbl 0764.30036 |

[6] | G. P. Scott, Compact submanifolds of 3-manifolds, J. London Math. Soc. (2) 7 (1973), 246 – 250. · Zbl 0266.57001 |

[7] | W. Thurston, The geometry and topology of 3-manifolds, lecture notes, Princeton Univ. |

[8] | Friedhelm Waldhausen, On irreducible 3-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56 – 88. · Zbl 0157.30603 |

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