By definition, a group is FC if all its conjugacy classes are finite.
Has anything been published about a generalization of the FC property for topological groups?
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2$\begingroup$ Yes, for locally compact groups check §2 in arxiv.org/abs/1306.4194 and references therein. $\endgroup$– YCorDec 12, 2019 at 14:32
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$\begingroup$ @YCor Thank you, that's very helpful. I'd be happy to accept this as an answer. At the same time, I am still curious if anyone has looked at other generalizations of the FC property, groups in which conjugacy classes are small in some other sense (meager, measure zero, ...). $\endgroup$– TJPDec 12, 2019 at 16:37
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1$\begingroup$ Most interesting non-discrete locally compact groups have conjugacy class whose closure has empty interior and in particular are meager of measure zero. This applies to all non-discrete locally compact abelian groups, all $\mathrm{GL}_n(K)$ or $\mathrm{SL}_n(K)$ for $K=\mathbf{R},\mathbf{C},\mathbf{Q}_p$, the direct product of any locally compact group with $\mathbf{R}$, etc. So it sounds hopeless to have an analogue of the results about FC-groups under such assumptions. $\endgroup$– YCorDec 12, 2019 at 16:42
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In 1963, Usakov characterised topological FC groups:
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1$\begingroup$ Also S.P. Wang, "Compactness properties of topological groups", Trans. AMS 154 (1971), 301--314. Also, arxiv.org/abs/1306.4194 and references therein, as pointed out YCor in his comment. In all these sources the topological FC property is the one where conjugacy classes are relatively compact. $\endgroup$– TJPDec 17, 2019 at 14:39