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S-Subgroups of the Real Hyperbolic Groups

Published online by Cambridge University Press:  20 November 2018

Thomas J. O'Malley
Thomas J. O'Malley*
Affiliation:
Le Moyne College, Syracuse, New York
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If H is a closed subgroup of a locally compact group G, with G/H having finite G-invariant measure, then, as observed by Atle Selberg [8], for any neighborhood U of the identity in G and any element g in G, there is an integer n > 0 such that gn is in U·H·U. A subgroup satisfying this latter condition is said to be an S-sub group, or satisfies property (S). If G is a solvable Lie group, then the converse of Selberg's result has been proved by S. P. Wang [10]: If H is a closed S-subgroup of G, then G/H is compact. Property (S) has been used by A. Borel in the important “density theorem” (see Section 2 or [1]).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 1 , 01 February 1980 , pp. 246 - 256
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Borel, A., Density properties for certain subgroups of semisimple groups without compact components, Ann. of Math. (2. 72 (1960), 179–188.Google Scholar
2. Cassels, J. W. S., An introduction to diophantine approximation (Cambridge Univ. Press, Cambridge, 1965).Google Scholar
3. Chen, S. S. and Greenberg, L., Hyperbolic spaces, in Contributions to analysis (Academic Press, New York, 1974).Google Scholar
4. Greenberg, L., Discrete subgroups of the Lorentz group, Math. Scand. 10 (1962), 85–107.Google Scholar
5. Greenberg, L., Commensurable groups of Moebius transformations, in Discontinuous groups and Riemann surfaces, Annals of Math. Studies, 79 (Princeton Univ. Press, Princeton, 1974), 227–237.Google Scholar
6. Hardy, G. H. and Wright, E. M., An Introduction to the theory of numbers (Oxford Univ. Press, London, 1960).Google Scholar
7. O'Malley, T. J., The geometry of S-subgroups of the hyperbolic groups, Ph.D. dissertation, Cornell University, 1974.Google Scholar
8. Selberg, A., On discontinuous groups in higher-dimensional symmetric spaces, in Contributions to function theory, Tata Institute, Bombay, 1960, 147–164.Google Scholar
9. Tukia, P., On discrete groups of the unit disk and their isomorphisms, Ann. Acad. Sci. Fenn., Series A.I., No. 504, 1972.Google Scholar
10. Wang, S. P., On S-subgroups of solvable Lie groups, Amer. J. Math. 92 (1970), 389–397.Google Scholar