Strong amenability and the infinite conjugacy class property. (English) Zbl 1429.37018
Let \(X\) be a compact Hausdorff space and \(G\) be a countable discrete group acting on \(X\). The authors show that \(G\) is strongly amenable (see [S. Glasner, Proximal flows. Berlin-Heidelberg-New York: Springer-Verlag (1976; Zbl 0322.54017)]) if and only if it has no quotients with infinite conjugacy class property (ICC). The groups that have no ICC quotients are also known as hyper-FC-central (see [D. H. McLain, Proc. Glasg. Math. Assoc. 3, 38–44 (1956; Zbl 0072.25702)]) or hyper-FC (see [A. M. Duguid, Pac. J. Math. 10, 117–120 (1960; Zbl 0102.26204)]).
As a corollary of such characterization the authors obtain that a finitely generated group is strongly amenable if and only if it is virtually nilpotent.
The proof of the main result relies on considering of ICC group \(G\) acting on some symbolic space and using genericity (in the Baire category sense) arguments.
As a corollary of such characterization the authors obtain that a finitely generated group is strongly amenable if and only if it is virtually nilpotent.
The proof of the main result relies on considering of ICC group \(G\) acting on some symbolic space and using genericity (in the Baire category sense) arguments.
Reviewer: Ivan Podvigin (Novosibirsk)
MSC:
| 37C85 | Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) |
| 37B02 | Dynamics in general topological spaces |
| 37B05 | Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) |
| 22D40 | Ergodic theory on groups |
| 22D25 | \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations |
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