|
On extendability of group actions on compact Riemann surfaces
Author(s):
Emilio
Bujalance;
F.
J.
Cirre;
Marston
Conder
Journal:
Trans. Amer. Math. Soc.
355
(2003),
1537-1557.
MSC (2000):
Primary 20H10;
Secondary 14H55, 20F38, 30F10
Posted:
December 4, 2002
Retrieve article in:
PDF DVI PostScript
Abstract |
References |
Similar articles |
Additional information
Abstract:
The question of whether a given group  which acts faithfully on a compact Riemann surface  of genus  is the full group of automorphisms of  (or some other such surface of the same genus) is considered. Conditions are derived for the extendability of the action of the group  in terms of a concrete partial presentation for  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  is a non-cyclic abelian group.
References:
-
- 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
 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:
10.1090/S0002-9947-02-03184-7
PII:
S 0002-9947(02)03184-7
Keywords:
Riemann surface,
automorphism group
Received by editor(s):
December 10, 2001
Posted:
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
Copyright of article:
Copyright
2002,
American Mathematical Society
|
Comments: webmaster@ams.org