Osaka Journal of Mathematics

Greenberg's theorem and equivalence problem on compact Riemann surfaces

Satoru Mizuta and Makoto Namba

Full-text: Open access

Abstract

Another proof of Greenberg's theorem on automorphism groups of compact \mboxRiemann surfaces is given. Using the idea of the proof, the equivalence problem for finite Galois coverings of the compact projective line is answered affirmatively, except special type of coverings.

Article information

Source
Osaka J. Math., Volume 43, Number 1 (2006), 137-178.

Dates
First available in Project Euclid: 28 April 2006

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1146242999

Mathematical Reviews number (MathSciNet)
MR2222406

Zentralblatt MATH identifier
1105.14039

Subjects
Primary: 14H37: Automorphisms
Secondary: 14H30: Coverings, fundamental group [See also 14E20, 14F35]

Citation

Mizuta, Satoru; Namba, Makoto. Greenberg's theorem and equivalence problem on compact Riemann surfaces. Osaka J. Math. 43 (2006), no. 1, 137--178. https://projecteuclid.org/euclid.ojm/1146242999


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References

  • A.F. Beardon: The Geometry of Discrete Groups, Springer-Verlag, 1983.
  • J.S. Birman: Braids, Links, and Mapping Class Groups, Ann. Math. Studies 82, Princeton, 1974.
  • R.H. Fox: Covering spaces with singularities: in Lefschetz Symposium, Princeton Univ. Press, 1957, 243--262.
  • H. Grauert and R. Remmert: Komplexe Räume, Math. Ann. 136 (1958), 245--318.
  • L. Greenberg: Maximal Fuchsian groups, Bull. Amer. Math. Soc. 69 (1963), 569--573.
  • Y. Imayoshi and M. Taniguchi: An Introduction to Teichmüller Spaces, Springer-Varlag, 1992.
  • G.A. Jones and D. Singerman: Complex Functions---an Algebraic and Geometric Viewpoint, Cambridge Univ. Press, 1987.
  • T. Kato: Conformal equivalence of compact Riemann surfaces, Japan J. Math. 7 (1981), 281--289.
  • F. Klein: Gesammelte Mathematische Abhandlungen, Dritter Band, Reprinted by Springer-Verlag, 1973.
  • T. Matsuno: On a theorem of Zariski-Vankampen type and its applications, Osaka J. Math. 32 (1995), 645--658.
  • D. Mumford, J. Fogarty and F. Kirwan: Geometric Invariant Theory, Third enlarged edition, Springer-Verlag, 1994.
  • M. Namba: Equivalence problem and automorphism groups of certain compact Riemann surfaces, Tukuba J. Math. 5 (1981), 319--338.
  • M. Namba: Families of Meromorphic Functions on Compact Riemann Surfaces, Lecture Notes in Math. 767, Springer-Verlag, 1979.
  • M. Namba: Branched Coverings and Algebraic Functions, Reseach Notes in Math. 161, Pitman-Longman, 1987.
  • G. Pourcin: Théoremè de Douady au-dessus de $S$, Ann. Scuola Norm. Sup. Pisa 23 (1969), 451--459.
  • K. Sakurai and M. Suzuki: Equivalence problem and automorphism groups of some Abelian branched coverings of the Riemann sphere, Mem. Kyushu Univ. 42 (1988), 145--152.
  • H. Schuster: Zur Theorie der Deformationen kompakter komplexer Räume, Inv. Math. 9 (1970), 284--294.
  • H. Völklein: Moduli spaces for covers of the Riemann sphere, Israel J. Math. 85 (1994), 407--430.