Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On extendability of group actions on compact Riemann surfaces


Authors: Emilio Bujalance, F. J. Cirre and Marston Conder
Journal: Trans. Amer. Math. Soc. 355 (2003), 1537-1557
MSC (2000): Primary 20H10; Secondary 14H55, 20F38, 30F10
DOI: https://doi.org/10.1090/S0002-9947-02-03184-7
Published electronically: December 4, 2002
MathSciNet review: 1946404
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The question of whether a given group $G$ which acts faithfully on a compact Riemann surface $X$ of genus $g\ge 2$ is the full group of automorphisms of $X$ (or some other such surface of the same genus) is considered. Conditions are derived for the extendability of the action of the group $G$ in terms of a concrete partial presentation for $G$associated with the relevant branching data, using Singerman's list of signatures of Fuchsian groups that are not finitely maximal. By way of illustration, the results are applied to the special case where $G$ is a non-cyclic abelian group.


References [Enhancements On Off] (What's this?)

  • 1. A. F. Beardon, The Geometry of Discrete Groups, Graduate Texts in Mathematics, 91, Springer, New York, 1983. MR 85d:22026
  • 2. W. Bosma, J. Cannon and C. Playoust, The MAGMA Algebra System I: The User Language, J. Symbolic Comput. 24 (1997), 235-265.
  • 3. E. Bujalance and M. Conder, On cyclic groups of automorphisms of Riemann surfaces, J. London Math. Soc. (2) 59 (1999), 573-584. MR 2000g:20098
  • 4. E. Bujalance, J. M. Gamboa and G. Gromadzki, The full automorphism groups of hyperelliptic Riemann surfaces, Manuscripta Math. 79 (1993), 267-282. MR 94f:20093
  • 5. F. J. Cirre, Automorphism groups of real algebraic curves which are double covers of the real projective plane, Manuscripta Math. 101 (2000), 495-512. MR 2001f:14055
  • 6. L. Greenberg, Maximal Fuchsian groups, Bull. Amer. Math. Soc. 69 (1963), 569-573. MR 26:6127
  • 7. L. Greenberg, Maximal groups and signatures, in Discontinuous groups and Riemann surfaces, Ann. of Math. Studies, No. 79, (Princeton, 1974), 207-226. MR 52:740
  • 8. W. J. Harvey, Cyclic groups of automorphisms of a compact Riemann surface, Quart. J. Math. 17 (1966), 86-97. MR 34:1511
  • 9. C. Maclachlan, Abelian groups of automorphisms of compact Riemann surfaces, Proc. London Math. Soc. (3) 15 (1965), 699-712. MR 31:3596
  • 10. D. Singerman, Finitely maximal Fuchsian groups. J. London Math. Soc. (2) 6 (1972), 29-38. MR 48:529
  • 11. D. Singerman and P. Watson, Non-maximal cyclic group actions on compact Riemann surfaces. Rev. Mat. Complutense (2) 10 (1997), 423-439. MR 99a:20050
  • 12. P. Turbek, An explicit family of curves with trivial automorphism groups. Proc. Amer. Math. Soc. 122 (1993), 657-664. MR 95a:14034
  • 13. P. Turbek, Sufficient conditions for a group of automorphisms of a Riemann surface to be its full automorphism group. J. Pure Appl. Algebra 123 (1998), 285-300. MR 99c:14045
  • 14. A. Wiman, Über die hyperelliptischen Kurven und diejenigen vom Geschlecht $p=3,$ welche eindeutige Transformationen in sich zulassen, Bihang Till. Kungl. Svenska Vetenskaps-Akademiens Handlingar 21, no. 1, (1895), 23 pp.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 20H10, 14H55, 20F38, 30F10

Retrieve articles in all journals with MSC (2000): 20H10, 14H55, 20F38, 30F10


Additional Information

Emilio Bujalance
Affiliation: Departamento de Matemáticas Fundamentales, Facultad de Ciencias, UNED, c/ Senda del Rey s/n, 28040 Madrid, Spain
Email: eb@mat.uned.es

F. J. Cirre
Affiliation: Departamento de Matemáticas Fundamentales, Facultad de Ciencias, UNED, c/ Senda del Rey s/n, 28040 Madrid, Spain
Email: jcirre@mat.uned.es

Marston Conder
Affiliation: Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand
Email: conder@math.auckland.ac.nz

DOI: https://doi.org/10.1090/S0002-9947-02-03184-7
Keywords: Riemann surface, automorphism group
Received by editor(s): December 10, 2001
Published electronically: December 4, 2002
Additional Notes: The first author was partially supported by DGICYT PB98-0017
The second author was partially supported by DGICYT PB98-0756
The third author was partially supported by N.Z. Marsden Fund UOA-810
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society