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On isomorphisms of geometrically finite Möbius groups. (English) Zbl 0572.30036
The goal of the paper is to prove that any type-preserving isomorphism \(j: G\to G'\) of geometrically finite Möbius groups G and G’ in \(\bar R^ n=R^ n\cup (\infty)\) (\(G,G'\) are discrete and have finite-sided fundamental polyhedra in the half-space \(R_+^{n+1})\) is induced by a unique homeomorphism \(f_ j: L(G)\to L(G')\) of the limit sets, i.e. \(f_ j(g(x))=j(g)(f_ j(x))\) for \(g\in G\) and \(x\in L(G)\). This result (Th. 3.3) is contained in the section 3. In his proof an important role is played by numerous statements on geometrically finite groups contained in section 2 (it takes nearly half of the paper).
Note that these results are greatly covered by the reviewer’s earlier papers [Sib. Mat. Zh. 23, No.6, 16-27 (1982; Zbl 0519.30038), Ann. Global Anal. Geom. 1, No.3, 1-22 (1983; Zbl 0531.57012)], see also the reviewer’s book ”Discrete transformation groups and manifold structures”, (Russian) (1983; Zbl 0571.57002).
Another important result (Th. 3.8) of the paper shows that if \(L(G)\neq \bar R^ n\) and if \(f: \bar R^ n-L(G)\to \bar R^ n-L(G')\) is a homeomorphism inducing j, then j is type-preserving if \(n\geq 2\) and that then f and \(f_ j\) define together a homeomorphism \(f'\) inducing j; for \(n\geq 2\) \(f'\) is quasiconformal if f is and the dilatation is not increased in the extension to \(L(G)\). In particular, for conformal f(and hence f’ is a Möbius transformation), this result is consistent with Mostow’s rigidity theorem for \(L(G)=\bar R^ n\) (see G. D. Mostow, ”Strong rigidity of locally symmetric spaces” (1973; Zbl 0265.53039), as was already observed by A. Marden [Ann. of Math., II. Ser. 99, 383- 462 (1974; Zbl 0282.30014)] for \(n=2\). Note that not only geometrically finite groups possess such rigidity. The exact class of rigidity in the groups of this sense in \(\bar R^ n\), \(n\geq 3\), is indicated by the reviewer [Analytic functions, Proc. Conf., Błażejewko/Pol. 1982, Lect. Notes Math. 1039, 1-8 (1983; Zbl 0527.57024); Dokl. Akad. Nauk SSSR 243, 829-832 (1978; Zbl 0414.20038)]. In the final section 4 the author examines for \(n=2\) when an isomorphism j is induced by a homeomorphism of \(\bar R_+^ 3\) (cf. A. Marden (loc. cit.; Th. 4.2)).
Reviewer: B.N.Apanasov

MSC:
30C62 Quasiconformal mappings in the complex plane
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
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