Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-24T22:41:05.750Z Has data issue: false hasContentIssue false
  • English
  • Français

On the order of automorphism groups of Klein surfaces

Published online by Cambridge University Press:  18 May 2009

J. J. Etayo Gordejuela
J. J. Etayo Gordejuela
Affiliation:
Departamento de Geometría y Topología, Facultad de Ciencias Matemáticas, Universidad Complutense, Madrid, Spain
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A problem of special interest in the study of automorphism groups of surfaces are the bounds of the orders of the groups as a function of the genus of the surface.

May has proved that a Klein surface with boundary of algebraic genus p has at most 12(p–1) automorphisms [9].

In this paper we study the highest possible prime order for a group of automorphisms of a Klein surface. This problem was solved for Riemann surfaces by Moore in [10]. We shall use his results for studying the Klein surfaces that are not Riemann surfaces. The more general result that we obtain is the following: if X is a Klein surface of algebraic genus p, and G is a group of automorphisms of X, of prime order n, then n ≤ p + 1.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 26 , Issue 1 , January 1985 , pp. 75 - 81
Copyright
Copyright © Glasgow Mathematical Journal Trust 1985

References

REFERENCES

1.Ailing, N. L., Real elliptic curves. Notes of Math. 54, (North-Holland, 1981).CrossRefGoogle Scholar
2.Ailing, N. L. and Greenleaf, N., Foundations of the theory of Klein surfaces. Lecture Notes in Math. 219, (Springer-Verlag, 1971).CrossRefGoogle Scholar
3.Bujalance, E., Normal subgroups of NEC groups, Math. Z. 178 (1981) 331–341.CrossRefGoogle Scholar
4.Bujalance, E., Cyclic groups of automorphisms of compact non-orientable Klein surfaces without boundary, Pacific J. of Math. 109 (1983) 279–289.CrossRefGoogle Scholar
5.Bujalance, E., Automorphisms groups of compact Klein surfaces with one boundary component, (to appear).Google Scholar
6.Bujalance, E. and Gamboa, J. M., Automorphisms groups of algebraic curves of ℝn of genus 2, Archiv der Math. 42 (1984) 229–237.CrossRefGoogle Scholar
7.Etayo, J. J., NEC subgroups in Klein surfaces. Bol. Soc. Mat. Mex. (to appear).Google Scholar
8.May, C. L., Large automorphism groups of compact Klein surfaces with boundary, Glasgow Math. J. 18 (1977) 1–10.CrossRefGoogle Scholar
9.May, C. L., A bound for the number of automorphisms of a compact Klein surface with boundary, Proc. Amer. Math. Soc. 63 (1977) 273–280.CrossRefGoogle Scholar
10.Moore, M. J., Fixed points of automorphisms of compact Riemann surfaces, Can. J. Math. 22 (1970) 922–932.CrossRefGoogle Scholar
11.Natanzon, S. M., Automorphisms of the Riemann surface of an M-curve, Fund Anal, and Appl. 12 (1978) 228–229.CrossRefGoogle Scholar
12.Preston, R., Projective structures and fundamental domains on compact Klein surfaces, (Ph.D. thesis. Univ. of Texas, 1975).Google Scholar
13.Singerman, D., On the structure of non-euclidean crystallographic groups, Proc. Cambridge Phil. Soc. 76 (1974) 233–240.CrossRefGoogle Scholar