Skip to main content Accessibility help
×
Home
Hostname: page-component-684899dbb8-plzwj Total loading time: 0.229 Render date: 2022-05-18T13:53:07.638Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true }

On the order of automorphism groups of Klein surfaces

Published online by Cambridge University Press:  18 May 2009

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

HTML view is 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 np + 1.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 1985

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) 331341.CrossRefGoogle Scholar
4.Bujalance, E., Cyclic groups of automorphisms of compact non-orientable Klein surfaces without boundary, Pacific J. of Math. 109 (1983) 279289.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) 229237.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) 110.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) 273280.CrossRefGoogle Scholar
10.Moore, M. J., Fixed points of automorphisms of compact Riemann surfaces, Can. J. Math. 22 (1970) 922932.CrossRefGoogle Scholar
11.Natanzon, S. M., Automorphisms of the Riemann surface of an M-curve, Fund Anal, and Appl. 12 (1978) 228229.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) 233240.CrossRefGoogle Scholar
You have Access
6
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

On the order of automorphism groups of Klein surfaces
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

On the order of automorphism groups of Klein surfaces
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

On the order of automorphism groups of Klein surfaces
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *