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



Abelian extremely amenable group?
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).


