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Strong amenability and the infinite conjugacy class property

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Abstract

A group is said to be strongly amenable if each of its proximal topological actions has a fixed point. We show that a finitely generated group is strongly amenable if and only if it is virtually nilpotent. More generally, a countable discrete group is strongly amenable if and only if none of its quotients have the infinite conjugacy class property.

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Notes

  1. Glasner attributes one of these examples to Furstenberg.

  2. For a self-contained proof see [6].

  3. A topological action \(G \curvearrowright X\) is strongly proximal if for each Borel probability measure \(\mu \) on X there exists a net \(\{g_i\}\) such that \(\lim _i g_i\mu \) is a point mass.

  4. Recall that given a finite symmetric subset \(X \subset G\) and \(g, h \in G\), we say that g and h are X-apart if \(g^{-1}h \not \in X\).

  5. The X-interiors of \(Y^{100}X\) and YX are \(Y^{100}\) and Y.

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Acknowledgements

We would like to thank Benjamin Weiss and Andrew Zucker for correcting mistakes in earlier drafts of this paper, and to likewise thank an anonymous referee for many corrections and suggestions. We would also like to thank Yair Hartman and Mehrdad Kalantar for drawing our attention to the relation of our results to the unique trace property of group von Neumann algebras.

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  1. California Institute of Technology, Pasadena, USA

    Joshua Frisch, Omer Tamuz & Pooya Vahidi Ferdowsi

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  1. Joshua Frisch
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  2. Omer Tamuz
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  3. Pooya Vahidi Ferdowsi
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Correspondence to Omer Tamuz.

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This work was supported by a grant from the Simons Foundation (#419427, Omer Tamuz), and by NSF Grant DMS-1464475.

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Frisch, J., Tamuz, O. & Vahidi Ferdowsi, P. Strong amenability and the infinite conjugacy class property. Invent. math. 218, 833–851 (2019). https://doi.org/10.1007/s00222-019-00896-z

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