Acta Mathematica

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

Quasiconformal extension of quasisymmetric mappings compatible with a Möbius group

  • Pekka Tukia


Quasisymmetric Mapping Quasiconformal Extension 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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.zbMATHMathSciNetGoogle Scholar
  2. [2]
    Ahlfors, L. V. &Weill, G., A uniqueness theorem for Beltrami equations.Proc. Amer. Math. Soc., 13 (1962), 975–978.CrossRefMathSciNetzbMATHGoogle 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.zbMATHMathSciNetGoogle Scholar
  7. [7]
    — Quasiconformal mappings, with applications to differential equations, function theory and topology.Bull. Amer. Math. Soc., 83 (1977), 1083–1100.zbMATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    Beurling, A. &Ahlfors, L. V., The boundary correspondence under qusiconformal mappings.Acta Math., 96 (1956), 125–142.MathSciNetzbMATHGoogle 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.zbMATHMathSciNetGoogle Scholar
  12. [12]
    Hempel, J., Residual finiteness of surface groups.Proc. Amer. Math. Soc., 32 (1972), 323.CrossRefzbMATHMathSciNetGoogle Scholar
  13. [13]
    Kra, I., On Teichmüller spaces for finitely generated Fuchsian groups.Amer. J. Math., 91 (1969), 67–74.zbMATHMathSciNetGoogle Scholar
  14. [14]
    Kühnau, R., Wertannahmeprobleme bei quasikonformen Abbildungen mit ortsabhängiger Dilatationsbeschränkung.Math. Nachr., 40 (1969), 1–11.zbMATHMathSciNetGoogle 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.zbMATHMathSciNetGoogle 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).zbMATHMathSciNetGoogle 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).zbMATHMathSciNetGoogle 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.zbMATHMathSciNetGoogle 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.zbMATHMathSciNetGoogle 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.zbMATHMathSciNetGoogle 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.zbMATHMathSciNetCrossRefGoogle 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.zbMATHMathSciNetGoogle Scholar
  35. [35]
    —, On infinite dimensional Techmüller spaces.Ann. Acad. Sci. Fenn. Ser. A I 3 (1977), 343–372.zbMATHMathSciNetGoogle Scholar
  36. [36]
    —, On two-dimensional quasiconformal groups.Ann. Acad. Sci. Fenn. Ser. A I 5 (1980), 73–78.zbMATHMathSciNetGoogle 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.zbMATHGoogle 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