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Is there a nontrivial commutative Hausdorff topological group that is extremely amenable?

Recall that a topological group is called extremely amenable if any continuous action on a compact Hausdorff topological space has a fixed point. For instance, it is known that no nontrivial locally compact group is extremely amenable, but some Polish groups, such as the group of order-preserving self-homeomorphisms of $[0,1]$, are extremely amenable.

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  • $\begingroup$ I added a definition of extremely amenable group, as I suppose that it is not something that everybody knows. I also tried to create a corresponding tag, but it is forgotten from mobile version (why?!) $\endgroup$ Jan 3 '16 at 15:51
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    $\begingroup$ @FedorPetrov I don't think such a specific tag is needed. $\endgroup$
    – YCor
    Jan 3 '16 at 16:06
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The very first examples of extremely amenable Polish groups were abelian, if I remember correctly (the so-called "exotic" groups of Herer and Christensen, Math. Ann. 213 (1975), 203-210).

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  • $\begingroup$ Yes, right. I was distracted by more recent more famous examples. $\endgroup$ Jan 4 '16 at 22:00
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Yes.

Typing "abelian extremely amenable" into Google gave me, on the first page, http://arxiv.org/abs/1201.0691

(Typing oniThing right now, will add context/details later)

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    $\begingroup$ Thank you Yemon, thank you Google. Alas, I may formally accept only one answer. $\endgroup$ Jan 4 '16 at 22:00

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