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A geometric proof of Mostow’s rigidity theorem for groups of divergence type. (English) Zbl 0532.30038
As 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 An earlier article by D. Sullivan, and subsequent (as yet unpublished) results by P. Tukia go beyond.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.

30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
Full Text: DOI
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