Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-26T02:26:52.616Z Has data issue: false hasContentIssue false
  • English
  • Français

Groups generated by elements with rational fixed points

Published online by Cambridge University Press:  20 January 2009

A. W. Mason
A. W. Mason
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow, G12 8QW, E-mail address:awm@MATHS.GLA.AC.UK
Rights & Permissions [Opens in a new window]

Abstract

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.

Let R be a commutative integral domain and let S be its quotient field. The group GL2(R) acts on Ŝ = S ∪ {∞} as a group of linear fractional transformations in the usual way. Let F2(R, z) be the stabilizer of z ∈ Ŝ in GL2(R) and let F2(R) be the subgroup generated by all F2(R, z). Among the subgroups contained in F2(R) are U2(R), the subgroup generated by all unipotent matrices, and NE2(R), the normal subgroup generated by all elementary matrices.

We prove a structure theorem for F2(R, z), when R is a Krull domain. A more precise version holds when R is a Dedekind domain. For a large class of arithmetic Dedekind domains it is known that the groups NE2(R),U2(R) and SL2(R) coincide. An example is given for which all these subgroups are distinct.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 40 , Issue 1 , February 1997 , pp. 19 - 30
Copyright
Copyright © Edinburgh Mathematical Society 1997

References

REFERENCES

1. Bourbaki, N., Commutative algebra (Hermann, Paris, 1972).Google Scholar
2. Cohn, P. M., On the structure of the GL 2of a ring, Publ. Math. I.H.E.S. 30 (1966), 365–413.CrossRefGoogle Scholar
3. Gilmer, R., Multiplicative Ideal Theory in Queen's Papers in Pure and Applied Mathematics (volume 90, Kingston, Ontario, Canada, 1992).Google Scholar
4. Grunewald, F., Mennicke, J. and Vaserstein, L., On the groups SL 2(Z[x]) and SL 2(k[x, y]), Israel J. Math. 86 (1994), 157–193.CrossRefGoogle Scholar
5. Liehl, B., On the group SL 2 over orders of arithmetic type, J. Reine Angew. Math. 323 (1981), 345–358.Google Scholar
6. Mason, A. W., Free quotients of congruence subgroups of SL 2 over a Dedekind ring of arithmetic type contained in a function field, Math. Proc. Cambridge Philos. Soc. 101 (1987), 421–429.CrossRefGoogle Scholar
7. Mason, A. W., Free quotients of the Bianchi groups and unipotent matrices, submitted for publication.Google Scholar
8. Mason, A. W., Free quotients of congruence subgroups of SL 2 over a coordinate ring and unipotent matrices, submitted for publication.Google Scholar
9. Mason, A. W., Unipotent matrices, modulo elementary matrices, in SL 2 over a coordinate ring, submitted for publication.Google Scholar
10. Radtke, W., Diskontinuierliche arithmetische Gruppen im Funktionenkörperfall, J. Reine Angew. Math. 363 (1985), 191–200.Google Scholar
11. Serre, J.-P., Le problème des groupes de congruence pour SL 2, Ann. of Math. 92 (1970), 489–527.CrossRefGoogle Scholar
12. Serre, J.-P., Trees (Springer-Verlag, Berlin, 1980).CrossRefGoogle Scholar
13. Stark, H. M., A complete determination of the complex quadratic fields of class-number one, Michigan Math. J. 14 (1967), 1–27.CrossRefGoogle Scholar
14. Suslin, A. A., One theorem of Cohn, J. Soviet Math. 17 (1981), 1801–1803.CrossRefGoogle Scholar
15. Vaserstein, L. N., On the group SL 2 over Dedekind rings of arithmetic type, Math. USSR – Sb. 18 (1972), 321–332.CrossRefGoogle Scholar