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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On centralizers of generalized uniform subgroups of locally compact groups

Author: Kwan-Yuk Law Sit
Journal: Trans. Amer. Math. Soc. 201 (1975), 133-146
MSC: Primary 22D05
MathSciNet review: 0354923
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Abstract: Let $ G$ be a locally compact group and $ H$ a closed subgroup of $ G$ such that the homogeneous space $ G/H$ admits a finite invariant measure. Let $ {Z_G}(H)$ be the centralizer of $ H$ in $ G$. It is shown that if $ G$ is connected then $ {Z_G}(H)$ modulo its center is compact. If $ G$ is only assumed to be locally connected it is shown that the commutator subgroup of $ {Z_G}(H)$ has compact closure. Consequences of these results are found for special classes of groups, such as Lie groups. An example of a totally disconnected group $ G$ is given to show that the results for $ {Z_G}(H)$ need not hold if $ G$ is not connected or locally connected.

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Keywords: Locally compact group, Lie group, periodic subset, compactness conditions, homogeneous space, invariant measure, Borel's density theorem, lattice
Article copyright: © Copyright 1975 American Mathematical Society