Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-15T11:38:55.657Z Has data issue: false hasContentIssue false
  • English
  • Français

Generic Dirichlet polygons and the modular group

Published online by Cambridge University Press:  18 May 2009

A. M. Macbeath
A. M. Macbeath
Affiliation:
Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh PA. 15260
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.

The concept of “marked polygon”, made explicit in this paper, is implicit in all studies of the relationships between the edges and vertices of a fundamental polygon for Fuchsian group, as well as in the topology of surfaces. Once the matching of the edges under the action of the group is known, one can deduce purely combinatorially the distribution of the vertices into equivalence classes, or cycles. Knowing a little more, the order of the rotation group fixing a vertex in each cycle, we can write down a presentation for the group.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 27 , October 1985 , pp. 129 - 141
Copyright
Copyright © Glasgow Mathematical Journal Trust 1985

References

REFERENCES

1.Beardon, A. F., Hyperbolic polygons and Fuchsian groups, J. London Math. Soc, 20 (1979), 247–254.CrossRefGoogle Scholar
2.Beardon, A. F., The Geometry of Discrete Groups (Springer, 1983).Google Scholar
3.Bers, L., Uniformization, moduli, and Kleinian Groups, Bull. London Math. Soc. 4 (1972) 257–300.CrossRefGoogle Scholar
4.Coxeter, H. S. M., Regular Polytopes (London, 1948).Google Scholar
5.Coxeter, H. S. M. and Moser, W. O. J., Generators and Relations for Discrete Groups (Springer 1957).CrossRefGoogle Scholar
6.Rham, G. de, Sur les polygones générateurs des groupes Fuchsiens. l'Enseignement Math 17 (1971), 49–62.Google Scholar
7.Fricke, R. und Klein, F., Vorlesungen über die Theorie der automorphen Funktionen (Teubner, 1897).Google Scholar
8.Omar, A. A. Hussein, On some permutation representations of (2, 3, n)-groups Ph.D. thesis, (Birmingham, England, 1979).Google Scholar
9.Lehner, J., Discontinuous groups and automorphic functions Amer. Math. Soc, 1964).Google Scholar
10.Macbeth, A. M. and Singerman, D., Spaces of subgroups and Teichmüller Space, Proc. London Math. Soc. (3) 31 (1975) 211–256.Google Scholar
11.Magnus, W., Non-euclidean tesselations and their Groups. (Academic Press, 1974).Google Scholar
12.Maskit, B., On Poincaré's theorem for fundamental polygons, Advances in Math. 7 (1971) 219–230.CrossRefGoogle Scholar
13.Millington, M. H., Cycloidal subgroups of the modular group, Proc. London Math. Soc. (3) 19 (1969) 164–176.Google Scholar
14.Poincaré, H., Oeuvres completes, t. II.Google Scholar
15.Siegel, C. L., Some Remarks on Discontinuous Groups. Annals of Math. 46 (1945) 708–718.Google Scholar