A geometric proof of Mostow’s rigidity theorem for groups of divergence type.

*(English)*Zbl 0532.30038As is well known, any Möbius transformation \(\gamma\) on \(R^ n\) (\(\gamma\in M(n))\) has a unique extension to \(R^{n+1}\) which fixes the upper half space \(U^{n+1}\). Moreover, a discrete Möbius group (\(\Gamma\subseteq M(n))\) acts, through this extension, discontinuously in \(U^{n+1}\), giving rise to a possibly branched covering \(\pi:U^{n+1}\to U^{n+1}/\Gamma ={\mathfrak M}_{\Gamma}.\) Mostow’s rigidity theorem for discrete Möbius groups asserts that under certain conditions on \(\Gamma\), a quasiconformal mapping \(f:R^ n\to R^ n\) which is compatible with \(\Gamma\), is already conformal (\(f\in M(n))\). Compatibility means that \(f{\mathbb{O}}\gamma {\mathbb{O}}f^{-1}\in M(n)\) for every \(\gamma\in \Gamma\), and arises in a natural way for the extension f to \(R^ n=\partial U^{n+1}\) of a mapping \^f:\(U^{n+1}\to U^{n+1}\) which is in turn induced by any homeomorphism \(\tilde f\) of \({\mathfrak M}_{\Gamma}\) onto any other \({\mathfrak M}_{\Gamma '}.\)

Mostow’s first proof assumed that \(n\geq 2\), and that \(\Gamma\) has finite covolume. This means that M(n)/\(\Gamma\) has finite invariant measure, and is equivalent to requiring that \({\mathfrak M}_{\Gamma}\) has finite hyperbolic volume. The present article extends the results to groups of divergence type, a classification which includes the finite covolume groups, but still requires that \(\Gamma\) be of the first kind. It is equivalent to the requirement that \(\Gamma\) acts ergodically on \(R^ n\times R^ n\) (using the stereographic identification of \(R^ n\) with the n-sphere \(S^ n\), and ordinary Hausdorff measure in \(S^ n)\), and is stronger than requiring only that \(\Gamma\) act ergodically in \(R^ n.\)

The author emphasizes that this result was not then, and is certainly not now the last word in the area of Mostow rigidity. An earlier article by D. Sullivan, and subsequent (as yet unpublished) results by P. Tukia go beyond. Of more interest is perhaps the extension of an early (1931) density theorem of P. J. Myrberg, and its connection to the Mostow rigidity question.

Mostow’s first proof assumed that \(n\geq 2\), and that \(\Gamma\) has finite covolume. This means that M(n)/\(\Gamma\) has finite invariant measure, and is equivalent to requiring that \({\mathfrak M}_{\Gamma}\) has finite hyperbolic volume. The present article extends the results to groups of divergence type, a classification which includes the finite covolume groups, but still requires that \(\Gamma\) be of the first kind. It is equivalent to the requirement that \(\Gamma\) acts ergodically on \(R^ n\times R^ n\) (using the stereographic identification of \(R^ n\) with the n-sphere \(S^ n\), and ordinary Hausdorff measure in \(S^ n)\), and is stronger than requiring only that \(\Gamma\) act ergodically in \(R^ n.\)

The author emphasizes that this result was not then, and is certainly not now the last word in the area of Mostow rigidity. An earlier article by D. Sullivan, and subsequent (as yet unpublished) results by P. Tukia go beyond. Of more interest is perhaps the extension of an early (1931) density theorem of P. J. Myrberg, and its connection to the Mostow rigidity question.

##### MSC:

30F35 | Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) |

Full Text:
DOI

##### References:

[1] | Agard, S.,Elementary properties of Möbius transformations in R n with applications to rigidity theory. University of Minnesota Mathematics Report 82-110. · JFM 42.0092.02 |

[2] | Ahlfors, L. V.,Möbius transformations in several variables. Ordway Professorship Lectures in Mathematics, University of Minnesota, Minneapolis (1981). · Zbl 0517.30001 |

[3] | Ahlfors, L. V., Two lectures in Kleinian groups, inProceedings of the Romanian-Finnish seminar on Teichmüller spaces and quasiconformal mappings, Brasov (1969), 60–64. |

[4] | Beurling, A. &Ahlfors, L. V., The boundary correspondence under quasiconformal mappings.Acta Math., 96 (1956), 125–142. · Zbl 0072.29602 · doi:10.1007/BF02392360 |

[5] | Gehring, F. W., The Carathéodory convergence theorem for quasiconformal mappings in space.Ann. Acad. Sci. Fenn. A.I., 336 (1963), 1–21. · Zbl 0136.38102 |

[6] | Kuusalo, T., Boundary mappings of geometric isomorphisms of Fuchsian groups.Ann. Acad. Sci. Fenn. A.I., 545 (1973), 1–6. · Zbl 0272.30023 |

[7] | Lehner, J.,Discontinuous groups and automorphic functions. AMS Mathematical Surveys VIII, Providence (1964). · Zbl 0178.42902 |

[8] | Mostow, G. D., Quasiconformal mappings inn-space and the rigidity of hyperbolic space forms.Institut des Hautes Études Scientifiques, Publications Mathematiques, 34 (1968), 53–104. · Zbl 0189.09402 · doi:10.1007/BF02684590 |

[9] | Mostow, G. D.,Strong rigidity of locally symmetric spaces. Annals of Math. Studies 78 (1973), Princeton. · Zbl 0265.53039 |

[10] | Myrberg, P. J., Ein Approximationssatz für die Fuchsschen Gruppen.Acta Math., 57 (1931), 389–409. · JFM 57.0450.02 · doi:10.1007/BF02403050 |

[11] | Sullivan, D., On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, inProc. Stony Brook conference on Riemann surfaces and Kleinian groups (1978). |

[12] | Tsuji, M.,Potential theory in modern function theory. Tokyo (1959). · Zbl 0087.28401 |

[13] | Gottschalk, W. H. & Hedlund, G. A.,Topological dynamics. AMS Colloquium Publications XXXVI, Providence (1955). · Zbl 0067.15204 |

[14] | Agard, S., Remarks on the boundary mapping for a Fuchsian group. To appear. · Zbl 0588.30049 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.