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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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##### References:
  L. V. Ahlfors, Fundamental polyhedrons and limit point sets of Kleinian groups,Proc. Nat. Acad. Sci. U.S.A. 55 (1966), 251–254. · Zbl 0132.30801 · doi:10.1073/pnas.55.2.251  B. Apanasov, Geometrically finite hyperbolic structures on manifolds,Ann. Global Analysis and Geometry. 1 (3) (1983), 1–22. · Zbl 0531.57012 · doi:10.1007/BF02329729  ———-, Geometrically finite groups of spatial transformations,Sibirskij Mat. J. 23 (6) (1983), 16–27 (Russian).  ———-,Discrete transformation groups and structures of manifolds, Novosibirsk, Nauka, 1983 (Russian).  A. F. Beardon andB. Maskit, Limit points of Kleinian groups and finite sided fundamental polyhedra,Acta Math. 132 (1974), 1–12. · Zbl 0277.30017 · doi:10.1007/BF02392106  V. A. Efremovič andE. S. Tihomirova, Equimorphisms of hyperbolic spaces,Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 1139–1144 (Russian).  W. J. Floyd, Group completions and limit sets of Kleinian groups,Inventiones Math. 57 (1980), 205–218. · Zbl 0428.20022 · doi:10.1007/BF01418926  F. W. Gehring andB. P. Palka, Quasiconformally homogeneous domains,J. Analyse Math. 30 (1976), 50–74. · Zbl 0349.30019 · doi:10.1007/BF02786713  W. Hurewicz andH. Wallman,Dimension theory, Princeton University Press, Princeton, 1948. · Zbl 0036.12501  W. Jaco,Lectures on three-manifold topology, CBMS Conference.43, American Mathematical Society, Providence, 1980. · Zbl 0433.57001  K. Johannson, Homotopy equivalences of 3-manifolds with boundaries,Lecture Notes in Mathematics.761, Springer-Verlag, Berlin-Heidelberg-New York, 1979. · Zbl 0412.57007  J. A. Kelingos, On the maximal dilatation of quasiconformal extensions,Ann. Acad. Sci. Fenn. Ser. A I. 478 (1971), 1–8. · Zbl 0212.10502  O. Lehto andK. I. Virtanen,Quasiconformal mappings in the plane, Springer-Verlag, Berlin-Heidelberg-New York, 1973. · Zbl 0267.30016  A. Marden, The geometry of finitely generated kleinian groups,Annals of Math. 99 (1974), 383–462. · Zbl 0282.30014 · doi:10.2307/1971059  ———-, Isomorphisms between fuchsian groups, inAdvances in complex function theory, ed. byW. Kirwan &L. Zalcman,Lecture Notes in Mathematics 505, Springer Verlag, Berlin-Heidelberg-New York, 1976, 56–78.  A. Marden andB. Maskit, On the isomorphism theorem for Kleinian groups,Inventiones Math. 51 (1979), 9–14. · Zbl 0399.30037 · doi:10.1007/BF01389909  G. A. Margulis, Isometry of closed manifolds of constant negative curvature with the same fundamental group,Dokl. Akad. Nauk SSSR. 192 (1979), 736–737 (=Soviet Math. Dokl. 11 (1979), 722–723).  B. Maskit, On boundaries of Teichmüller spaces and on kleinian groups: II,Annals of Math. 91 (1970), 607–639. · Zbl 0197.06003 · doi:10.2307/1970640  ———-, Intersections of component subgroups of Kleinian groups, inDiscontinuous groups and Riemann surfaces, ed. byL. Greenberg,Annals of Mathematics Studies.79, Princeton University Press, Princeton, 1974, 349–367.  ———-, Isomorphisms of function groups,J. Analyse Math. 32 (1977), 63–82. · Zbl 0392.30028 · doi:10.1007/BF02803575  ———-, On the classification of Kleinian groups II-Signatures,Acta Math. 138 (1977), 17–42. · Zbl 0358.30011 · doi:10.1007/BF02392312  G. D. Mostow, Quasi-conformal mappings inn-space and the rigidity of hyperbolic space forms,Inst. Hautes Études Sci. Publ. Math. 34 (1968), 53–104. · Zbl 0189.09402 · doi:10.1007/BF02684590  ———-, Strong rigidity of locally symmetric spaces,Annals of Mathematics Studies.78, Princeton University Press, Princeton, 1973. · Zbl 0265.53039  J. Nielsen, Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen,Acta Math. 50 (1927), 189–358. · JFM 53.0545.12 · doi:10.1007/BF02421324  G. Prasad, Strong rigidity of Q-rank 1 lattices,Inventiones Math. 21 (1973), 255–286. · Zbl 0264.22009 · doi:10.1007/BF01418789  H. M. Reimann, Invariant extension of quasiconformal deformations, to appear. · Zbl 0592.30025  A. Selberg, On discontinuous groups in higher-dimensional symmetric spaces, inContributions to function theory, TATA Inst. of Fund. Research, Bombay, 1960, 147–164.  D. Sullivan, Hyperbolic geometry and homeomorphisms, inGeometric topology, Proceedings of the 1977 Georgia Topology Conference, ed. byJ. C. Cantrell, Academic Press, New York-London, 1979, 543–555.  W. P. Thurston,The geometry and topology of three-manifolds, Mimeographed lecture notes, Princeton, 1980.  P. Tukia, On discrete groups of the unit disk and their isomorphisms,Ann. Acad. Sci. Fenn. Ser. A I. 504 (1972), 1–45. · Zbl 0225.30022  ———-, Extension of boundary homeomorphisms of discrete groups of the unit disk, Ibid.548 (1973), 1–16.  —–, Quasiconformal extension of quasisymmetric mappings compatible with a Möbius group,Acta Math. 154 (1985). · Zbl 0562.30018  ———-, The Hausdorff dimension of the limit set of a geometrically finite Kleinian group,Acta Math. 152 (1984), 127–140. · Zbl 0539.30034 · doi:10.1007/BF02392194  —–, Rigidity theorems for Möbius groups, to appear.  —–, Differentiability and rigidity of Möbius groups, to appear. · Zbl 0564.30033  —–, On limit sets of geometrically finite Möbius groups,Math. Scandinavica, to appear.  P. Tukia andJ. Väisälä, Quasisymmetric embeddings of metric spaces,Ann. Acad. Sci. Fenn. Ser. A I. 5 (1980) 97–114. · Zbl 0403.54005  ———- ———-, Lipschitz and quasiconformal approximation and extension, Ibid.6 (1981), 303–342. · Zbl 0448.30021  ———- ———-, Quasiconformal extension from dimensionn ton + 1,Annals of Math. 115 (1982), 331–348. · Zbl 0484.30017 · doi:10.2307/1971394  J. Väisälä, Lectures onn-dimensional quasiconformal mappings,Lecture Notes in Mathematics.229, Springer-Verlag, Berlin-Heidelberg-New York, 1971.  N. J. Wielenberg, Discrete Moebius groups: Fundamental polyhedra and convergence,American J. of Math. 99 (1977), 861–877. · Zbl 0373.57024 · doi:10.2307/2373869  J. A. Wolf,Spaces of constant curvature, Publish or Perish Inc., Berkeley, 1977. · Zbl 0373.57025
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