# zbMATH — the first resource for mathematics

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)
Full Text: