Mostow rigidity on the line: A survey

  • Stephen Agard
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 11)

Abstract

G.D. Mostow’s celebrated Rigidity Theorem has taken some curious forms on the real line. All assume that f (a continuous strictly increasing real valued function of a real variable) is the “boundary mapping” of an isomorphism between Fuchsian groups.

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References

  1. [Agl]
    Agard, S., A geometric proof of Mostow’s rigidity theorem for groups of divergence type, Acta. Math. 151 (1983), 231–252.MathSciNetCrossRefMATHGoogle Scholar
  2. [Ag2]
    Agard, S., Elementary properties of Möbius transformations in Rn with applications to rigidity theory. University of Minnesota Math. Report 82–110, (1982), Minneapolis.Google Scholar
  3. [Ag3]
    Agard, S., Remarks on the boundary mapping for a Fuchsian group ,Ann. Acad. Sci. Fenn. A.I. Math 10 (1985), 1–13.MathSciNetMATHGoogle Scholar
  4. [Ahl]
    Ahlfors, L., “Möbius Transformations in Several Variables,” Ordway Professorship Lectures in Mathematics, University of Minnesota, Minneapolis, 1981.Google Scholar
  5. [Ah2]
    Ahlfors, L., Ergodic properties of Möbius transformations ,Analytic functions Kozub-nik 1979, Lecture Notes in Math. 798; Springer (1980), 1–9.Google Scholar
  6. [A/S]
    Ahlfors, L., and Sario, L., “Riemann Surfaces,” Princeton, 1960.MATHGoogle Scholar
  7. [B/M]
    Beardon, A.F., and Maskit, B., Limit points of Kleinian groups and finite sided fundamental polyhedra ,Acta. Math. 132 (1974), 1–12.MathSciNetCrossRefMATHGoogle Scholar
  8. [Bo]
    Bowen, R., Hausdorff dimension of quasicircles ,Inst. Hautes Etudes Sci. Publ. Math 50 (1979), 11–25.MathSciNetCrossRefMATHGoogle Scholar
  9. [C/C]
    Constantinescu, C., and Cornea, A., “Ideale Ränden Riemannschen Flächen,” Springer, 1963.Google Scholar
  10. [H]
    Hopf, E., Ergodic theory and the geodesic flow on surfaces of constant negative curvature, Bull. Amer. Math. Soc. 77 (1971), 863–877.MathSciNetCrossRefMATHGoogle Scholar
  11. [K]
    Kuusalo, T., Boundary mappings of geometric isomorphisms of Fuchsian groups ,Ann. Acad. Sci. Fenn. A.I. Math 545 (1973), 1–7.Google Scholar
  12. [K/T]
    Kusunoki, Y., and Taniguchi, M., Remarks on Fuchsian groups associated with open Riemann surfaces ,Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook conference, edited by I. Kra and B. Maskit, Ann. of Math. Studies 97 (1981); Princeton, 377–390.Google Scholar
  13. [Ml]
    Mostow, G.D., Quasiconformal mappings inn-space and the rigidity of hyperbolic space forms ,Inst. Hautes Etudes Sci. Publ. Math. 34 (1968), 53–104.MathSciNetCrossRefMATHGoogle Scholar
  14. [M2]
    Mostow, G.D., “Strong Rigidity of Locally Symmetric Spaces,” Ann. Math. Studies 78 (1973), Princeton.Google Scholar
  15. [S1]
    Sullivan, D., On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions ,Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook conference, edited by I. Kra and B. Maskit, Ann. of Math. Studies 97 (1981); Princeton, 465–496.Google Scholar
  16. [S2]
    Sullivan, D., Discrete conformal groups and measurable dynamics ,Bulletin (New Series) AMS 6 (No. 1) (1982), 57–73.CrossRefMATHGoogle Scholar
  17. [T]
    Tukia, P., Differentiability and rigidity of Möbius groups ,Invent. Math. 82 (1985); No. 3, 557–578.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

  • Stephen Agard
    • 1
  1. 1.University of MinnesotaMinneapolisUSA

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