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Note on a theorem of Weil-Abels

Part of: Lie groups

Published online by Cambridge University Press:  26 February 2010

N. Oler
N. Oler
Affiliation:
University of Pennsylvania, University of Birmingham.
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Extract

Let G and H be Hausdorff locally compact groups. By R(G, H) we denote the space of continuous homomorphisms of G into H equipped with the compact-open topology, namely that which is generated by subsets of the form

where K is any compact subset of G and U is any open subset of H. Further, let R0(G, H) be the subset of R(G, H) consisting of elements r satisfying the conditions that the quotient space H/r(G) is compact and r is proper, i.e., the action of G on H defined as the action by left translations of r(G) on H is proper. It has been shown by H. Abels [2] that if G and H are such that at least one of them has a compact defining subset then R0(G,H) is open in R(G,H). Moreover, for each r0 ∈ R0(G,H) there exists a neighbourhood M and a compact subset F of H such that r(G) F = H for all r ∈ M and for each compact subset K of H the union is a relatively compact subset of G. It is furthermore shown in [1] that if G contains no small subgroups and H is connected then the subset of R0(G, H) consisting of isomorphisms of G into H is open in R(G, H). These results, in the case in which H is a connected Lie Group and G is a discrete group, have been established by Weil in [5] and [6] appendix 1.

MSC classification

Secondary: 22E40: Discrete subgroups of Lie groups
Type
Research Article
Information
Mathematika , Volume 18 , Issue 2 , December 1971 , pp. 185 - 187
Copyright
Copyright © University College London 1971

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References

1.Abels, H., “Über die Erzeugung von eigentlichen Transformationsgruppen ”, Math. Z., 103 (1968), 333–357.CrossRefGoogle Scholar
2.Abels, H., “Über eigentliche Transformationsgruppen “, Math. Z., 110 (1969), 75–100.CrossRefGoogle Scholar
3.Bourbaki, N., Eléments de Mathématique, Intégration Chap. 8, Convolution et Représentations (Paris, Hermann, 1963).Google Scholar
4.Montgomery, D. and Zippin, L., Topological transformation groups, Interscience tracts (New York, 1955).Google Scholar
5.Weil, A., “On discrete subgroups of Lie groups I ”, Ann. of Math., 72 (1960), 369–384.CrossRefGoogle Scholar
6.Weil, A., “On discrete subgroups of Lie groups II ”, Ann. of Math., 75 (1962), 578–602.CrossRefGoogle Scholar