Extreme amenability of abelian L0 groups

We show that for any abelian topological group G and arbitrary diffused submeasure μ, every continuous action of L0(μ,G) on a compact space has a fixed point. This generalizes earlier results of Herer and Christensen, Glasner, Furstenberg and Weiss, and Farah and Solecki. This also answers a question posed by Farah and Solecki. In particular, it implies that if H is of the form L0(μ,R), then H is extremely amenable if and only if H has no nontrivial characters. The latter gives an evidence for an affirmative answer to a question of Pestov. The proof is based on estimates of chromatic numbers of certain graphs on Zn. It uses tools from algebraic topology and builds on the work of Farah and Solecki.