Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-19T00:47:59.660Z Has data issue: false hasContentIssue false
  • English
  • Français

On the uniqueness of arithmetic structures

Published online by Cambridge University Press:  14 November 2011

F. E. A. Johnson
F. E. A. Johnson
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E6BT, U. K.
Get access Rights & Permissions [Opens in a new window]

Extract

Margulis has given conditions under which a lattice in a semisimple Lie group admits the structure of an arithmetic subgroup. We show that these arithmetic structures are unique. The result is not subject to the condition “rkR(G) ≧ 2” required by the Margulis result. In the lowest dimensions, the result has previously been observed by Takeuchi, Maclachlan and Reid.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 5 , 1994 , pp. 1037 - 1044
Copyright
Copyright © Royal Society of Edinburgh 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Borel, A.. Density and maximality of arithmetic subgroups. J. Reine Angew. Math. 224 (1966), 78–89.Google Scholar
2Borel, A. and Arithmetic, Harish-Chandrasubgroups of algebraic groups. Ann. of Math. 75 (1962), 485–535.Google Scholar
3Borel, A. and Tits, J.. Homomorphismes “abstraits” de groupes algebriques simples. Ann. of Math. 97 (1973), 499–571.CrossRefGoogle Scholar
4Chevalley, C.. On algebraic group varieties. J. Math. Soc. Japan 6 (1954), 303–324.Google Scholar
5Freudenthal, H.. Die Topologie der Lieschen Gruppen als algebraisches Phänomen I. Ann. of Math. 42(1941), 1051–1074.Google Scholar
6Johnson, F. E. A.On the existence of irreducible discrete subgroups in isotypic Lie groups of classical type. Proc. London Math. Soc. 56 (1988), 51–77.CrossRefGoogle Scholar
7Maclachlan, C.. Fuchsian subgroups of the groups PSL2(Od). In Low Dimensional Topology and Kleinian Groups, ed. Epstein, D. B. A., Lecture, L. M. S. Notes 112, 305–311 (London: London Mathematical Society, 1986).Google Scholar
8Maclachlan, C. and Reid, A. W.. Commensurability classes of arithmetic Kleinian groups and their Fuchsian subgroups. Math. Proc. Cambridge Philos. Soc. 102 (1987), 251–258.CrossRefGoogle Scholar
9Margulis, G. A.. Discrete subgroups of semisimple Lie groups. Ergebnisse der Mathematik, 3 Folge, Bd. 17 (Berlin: Springer, 1991).Google Scholar
10Mostow, G. D.. Self-adjoint groups. Ann. of Math. 62 (1955), 44–55.Google Scholar
11Mostow, G. D. and Tamagawa, T.. On the compactness of arithmetically defined homogeneous spaces. Ann. of Math. 76 (1962), 446–463.CrossRefGoogle Scholar
12Raghunathan, M. S.. Discrete subgroups of Lie groups. Ergebnisse der Mathematik 68 (Berlin: Springer, 1972).Google Scholar
13Reid, A. W.. (Ph.D. Thesis, University of Aberdeen, 1987).Google Scholar
14Rosenlicht, M.. Some rationality questions on algebraic groups. Annali di Matematici 43 (1957), 25–50.CrossRefGoogle Scholar
15Takeuchi, K.. A characterisation of arithmetic Fuchsian groups. J. Math. Soc. Japan 27 (1975), 600–612.CrossRefGoogle Scholar
16St, D. J.. Webber, H.. On the existence of irreducible discrete subgroups in isotypic Lie groups of exceptional type (Ph.D. Thesis, University of London, 1985).Google Scholar
17Weil, A.. The field of definition of a variety. Amer. J. Math. 78 (1956), 509–524.CrossRefGoogle Scholar