Hostname: page-component-5b777bbd6c-cp4x8 Total loading time: 0 Render date: 2025-06-23T15:52:54.274Z Has data issue: false hasContentIssue false
  • English
  • Français

Discrete Groups of Motions

Published online by Cambridge University Press:  20 November 2018

Leon Greenberg
Leon Greenberg*
Affiliation:
Brown University
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.

This paper deals with the discrete groups of rigid motions of the hyperbolic plane. It is known (12) that the finitely generated, orientation-preserving groups have the following presentations:

Generators: .

Defining relations:

where km = ambmam-1bm-1. We shall denote this group by F(p; n1, … , nd; r).

In particular, the finitely generated free groups are contained among these. Indeed, one purpose of this paper is to indicate some geometrical methods for investigating free groups.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 415 - 426
Copyright
Copyright © Canadian Mathematical Society 1960

References

1. Bundgaard, S. and Nielsen, J., On normal subgroups with finite index in F-groups, Mat. Tids. B (1951), 56–58.Google Scholar
2. Coxeter, H.S.M., On subgroups of the modular group, J. de Math. Pures et App. (1958), 317–319.Google Scholar
3. Coxeter, H.S.M. and Moser, W.O.J., Generators and relations for discrete groups, Ergeb. der Math. (1957).Google Scholar
4. Ford, L., Automorphic functions (New York, 1951).Google Scholar
5. Fox, R., On Fenchel's conjecture about F-groups, Mat. Tids. B (1952), 61–65.Google Scholar
6. Goldberg, K., Unimodular matrices or order 2 that commute, J. Washington Acad. Sci., 46 (1956), 337–338.Google Scholar
7. Howson, A.G., On the intersection of finitely generated free groups, J. London Math. Soc, 29 (1954), 428–434.Google Scholar
8. Karrass, A. and Solitar, D., Note on a theorem of Schreier, Proc. Amer. Math. Soc, 8 (1957), 696–697.Google Scholar
9. Magnus, W., Discrete groups (New York University Notes, 1952).Google Scholar
10. Nielsen, J., Ueber Gruppen linearer Transformationen, Mitteilungen der Math. Ges. in Hamburg Band VIII (1940), 82–104.Google Scholar
11. Nielsen, J., Kommutatorgruppen for det frie product af cykliske grupper, Mat. Tids. B (1948), 49–56.Google Scholar
12. Nielsen, J., Nogle grundlaeggende begreber vedrϕrende diskontinuerte grupper af lineaere substitutioner i en kompleks variabel, Den IIte Skandinaviske Matematikerkongress i Trondheim (1949), 61–70.Google Scholar