Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-25T06:08:57.054Z Has data issue: false hasContentIssue false
  • English
  • Français

Automorphisms of compact non-orientable Riemann surfaces

Published online by Cambridge University Press:  18 May 2009

D. Singerman
D. Singerman
Affiliation:
University of Birmingham
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.

Using the definition of a Riemann surface, as given for example by Ahlfors and Sario, one can prove that all Riemann surfaces are orientable. However by modifying their definition one can obtain structures on non-orientable surfaces. In fact nonorientable Riemann surfaces have been considered by Klein and Teichmüller amongst others. The problem we consider here is to look for the largest possible groups of automorphisms of compact non-orientable Riemann surfaces and we find that this throws light on the corresponding problem for orientable Riemann surfaces, which was first considered by Hurwitz [1]. He showed that the order of a group of automorphisms of a compact orientable Riemann surface of genus g cannot be bigger than 84(g – 1). This bound he knew to be attained because Klein had exhibited a surface of genus 3 which admitted PSL (2, 7) as its automorphism group, and the order of PSL(2, 7) is 168 = 84(3–1). More recently Macbeath [5, 3] and Lehner and Newman [2] have found infinite families of compact orientable surfaces for which the Hurwitz bound is attained, and in this paper we shall exhibit some new families.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 12 , Issue 1 , April 1971 , pp. 50 - 59
Copyright
Copyright © Glasgow Mathematical Journal Trust 1971

References

REFERENCES

1.Hurwitz, A., Uber algebraische Gebilde mit eindeutigen Transformationen in sich, Math. Ann. 41 (1879), 428–471.Google Scholar
2.Lehner, J. and Newman, M., On Riemann surfaces with maximal automorphism groups, Glasgow Math. J. 8 (1967), 102–112.CrossRefGoogle Scholar
3.Macbeath, A. M., On a theorem of Hurwitz, Proc. Glasgow Math. Assoc. 5 (1961), 90–96.CrossRefGoogle Scholar
4.Macbeath, A. M., Discontinuous groups and birational transformations, Dundee Summer School (1961).Google Scholar
5.Macbeath, A. M., Generators of the linear fractional groups, Proc. Symp. Pure Math. (Houston, 1967).Google Scholar
6.Newman, M., Maximal normal subgroups of the modular group, Proc. Amer. Math. Soc. 19 (1968), 1138–1144.CrossRefGoogle Scholar
7.Wilkie, H. C., On non-Euclidean crystallographic groups, Math. Z. 91 (1966), 87–102.CrossRefGoogle Scholar