Acta Mathematica

, Volume 154, Issue 3–4, pp 153–193 | Cite as

Quasiconformal extension of quasisymmetric mappings compatible with a Möbius group

  • Pekka Tukia
Article

Keywords

Quasisymmetric Mapping Quasiconformal Extension 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Ahlfors, L. V., Extension of quasiconformal mappings from two to three dimensions.Proc. Nat. Acad. Sci. U.S.A., 51 (1964), 768–771.MATHMathSciNetGoogle Scholar
  2. [2]
    Ahlfors, L. V. &Weill, G., A uniqueness theorem for Beltrami equations.Proc. Amer. Math. Soc., 13 (1962), 975–978.CrossRefMathSciNetMATHGoogle Scholar
  3. [3]
    Beardon, A. F.,The geometry of discrete groups. Graduate Texts in Mathematics 91, Springer-Verlag, 1983.Google Scholar
  4. [4]
    Bers, L., Extremal quasiconformal mappings, inAdvances in the theory of Riemann surfaces. Ed. by L. V. Ahlfors et al. Annals of Mathematics Studies 66, Princeton University Press, 1971, 27–52.Google Scholar
  5. [5]
    Bers, L. Universal Teichmüller space, inAnalytic methods in mathematical physics. Ed. by R. P. Gilbert and R. G. Newton. Gordon and Breach, 1968, 65–83.Google Scholar
  6. [6]
    — Uniformization, moduli, and Kleinian groups.Bull. London Math. Soc., 4 (1972), 257–300.MATHMathSciNetGoogle Scholar
  7. [7]
    — Quasiconformal mappings, with applications to differential equations, function theory and topology.Bull. Amer. Math. Soc., 83 (1977), 1083–1100.MATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    Beurling, A. &Ahlfors, L. V., The boundary correspondence under qusiconformal mappings.Acta Math., 96 (1956), 125–142.MathSciNetMATHGoogle Scholar
  9. [9]
    Dold, A.,Lectures on algebraic topology. Springer-Verlag, 1972.Google Scholar
  10. [10]
    Efremovic, V. A. &Tihomirova, E. S., Equimorphisms of hyperbolic spaces.Izv. Akad. Nauk SSSR Ser. Math., 28 (1964), 1139–1144 (russian).MathSciNetGoogle Scholar
  11. [11]
    Greenberg, L., Discrete subgroups of the Lorentz group.Math. Scand., 10 (1962), 85–107.MATHMathSciNetGoogle Scholar
  12. [12]
    Hempel, J., Residual finiteness of surface groups.Proc. Amer. Math. Soc., 32 (1972), 323.CrossRefMATHMathSciNetGoogle Scholar
  13. [13]
    Kra, I., On Teichmüller spaces for finitely generated Fuchsian groups.Amer. J. Math., 91 (1969), 67–74.MATHMathSciNetGoogle Scholar
  14. [14]
    Kühnau, R., Wertannahmeprobleme bei quasikonformen Abbildungen mit ortsabhängiger Dilatationsbeschränkung.Math. Nachr., 40 (1969), 1–11.MATHMathSciNetGoogle Scholar
  15. [15]
    Kuusalo, T., Boundary mappings of geometric isomorphisms of Fuchsian groups.Ann. Acad. Sci. Fenn. Ser. A I, 545 (1973), 1–7.MathSciNetGoogle Scholar
  16. [16]
    Lehtinen, M., A real-analytic quasiconformal extension of a quasisymmetric function.Ann. Acad. Sci. Fenn. Ser. A I, 3 (1977), 207–213.MATHMathSciNetGoogle Scholar
  17. [17]
    Lehto, O., Group isomorphisms induced by qusiconformal mappings, inContributions to analysis. Ed. by L. V. Ahlfors et al. Academic Press, 1974, 241–244.Google Scholar
  18. [18]
    Lehto, O. & Virtanen, K. I.,Quasiconformal mappings in the plane. Springer-Verlag, 1973.Google Scholar
  19. [19]
    Macbeath, A. M., The classification of non-euclidean plane crystallographic groups.Canad. J. Math., 6 (1967), 1192–1205.MathSciNetGoogle Scholar
  20. [20]
    Malcev, A., On isomorphic matrix representations of infinite groups.Mat. Sb., 8 (50) (1940), 405–442 (russian).MATHMathSciNetGoogle Scholar
  21. [21]
    Marden, A., Isomorphisms between Fuchsian groups, inAdvances in complex function theory. Ed. by W. E. Kirwan and L. Zalcman. Lecture Notes in Mathematics 505, Springer-Verlag, 1976, 56–78.Google Scholar
  22. [22]
    Margulis, G. A., Isometry of closed manifolds of constant negative curvature with the same fundamental groups.Dokl. Akad. Nauk SSSR, 192 (1970), 736–737 (=Soviet Math. Dokl.) 11 (1970), 722–723).MATHMathSciNetGoogle Scholar
  23. [23]
    Martio, O., Rickman, S. &Väisälä, J., Distortion and singularities of quasiregular mappings.Ann. Acad. Sci. Fenn. Ser. A I 465 (1970), 1–13.Google Scholar
  24. [24]
    Mostow, G. D., Quasi-conformal mappings inn-space and the rigidity of hyperbolic space forms.Inst. Hautes Études Sci. Publ. Math., 34 (1968), 53–104.MATHMathSciNetGoogle Scholar
  25. [25]
    Mostow, G. D.,Strong rigidity of locally symmetric spaces. Annals of Mathematics Studies 78, Princeton University Press, 1973.Google Scholar
  26. [26]
    Nielsen, J., Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen.Acta Math., 50 (1927), 189–358.MATHMathSciNetGoogle Scholar
  27. [27]
    Rado, T. & Reichelderfer, P. V.,Continuous transformations in analysis. Springer-Verlag, 1955.Google Scholar
  28. [28]
    Sakan, K., On quasiconformal mappings compatible with a Fuchsian group.Osaka J. Math., 19 (1982), 159–170.MATHMathSciNetGoogle Scholar
  29. [29]
    Selberg, A., On discontinuous groups in higher-dimensional symmetric spaces, inContributions to function theory (Internat. Colloq. Function theory). Tata Institute of Fundamental Research, 1960, 147–164.Google Scholar
  30. [30]
    Strebel, K., On lifts of quasiconformal mappings.J. Analyse Math., 31 (1977), 191–203.MATHMathSciNetCrossRefGoogle Scholar
  31. [31]
    Thurston, W. P.,The geometry and topology of 3-manifolds. Mimeographed lecture notes, Princeton University, 1980.Google Scholar
  32. [32]
    Tukia, P., On discrete groups of the unit disk and their isomorphism.Ann. Acad. Sci. Fenn. Ser. A I, 504 (1972), 1–45.MathSciNetGoogle Scholar
  33. [33]
    — Extension of boundary homeomorphisms of discrete groups of the unit disk.Ann. Acad. Sci. Fenn. Ser. A I, 548 (1973), 1–16.MathSciNetGoogle Scholar
  34. [34]
    —, The space of quasisymmetric mappings.Math. Scand., 40 (1977), 127–142.MATHMathSciNetGoogle Scholar
  35. [35]
    —, On infinite dimensional Techmüller spaces.Ann. Acad. Sci. Fenn. Ser. A I 3 (1977), 343–372.MATHMathSciNetGoogle Scholar
  36. [36]
    —, On two-dimensional quasiconformal groups.Ann. Acad. Sci. Fenn. Ser. A I 5 (1980), 73–78.MATHMathSciNetGoogle Scholar
  37. [37]
    Tukia, P. Oh isomorphisms of geometrically finite Möbius groups. To appear inInst. Hautes Études Sci. Publ. Math. Google Scholar
  38. [38]
    Tukia, P., Differentiability and rigidity of Möbius groups. To appear.Google Scholar
  39. [39]
    Tukia, P., Automorphic quasimeromorphic mappings for torsionless hyperbolic groups.Ann. Acad. Sci. Fenn. Ser. A I, 10 (1985).Google Scholar
  40. [40]
    Tukia, P. &Väisälä, J., Quasisymmetric embeddings of metric spaces.Ann. Acad. Sci. Fenn. Ser. A I, 5 (1980), 97–114.MATHGoogle Scholar
  41. [41]
    —, Quasiconformal extension from dimensionn ton+1.Ann. of Math., 115 (1982), 331–348.CrossRefMathSciNetGoogle Scholar
  42. [42]
    Väisälä, J.,Lectures on n-dimensional quasiconformal mappings. Lecture Notes in Mathematics 229, Springer-Verlag, 1971.Google Scholar
  43. [43]
    Väisälä, J. Quasimöbius maps. To appear.Google Scholar
  44. [44]
    Zieschang, H., Vogt, E. & Coldewey, H.-D.,Surfaces and planar discontinuous groups. Lecture Notes in Mathematics 835, Springer-Verlag, 1980.Google Scholar

Copyright information

© Almqvist & Wiksell 1985

Authors and Affiliations

  • Pekka Tukia
    • 1
  1. 1.University of HelsinkiHelsinkiFinland

Personalised recommendations