# Abelian extremely amenable group?

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.

• 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?!) Jan 3 '16 at 15:51
• @FedorPetrov I don't think such a specific tag is needed.
– YCor
Jan 3 '16 at 16:06