I will discuss dynamical properties of large abelian groups. A topological group is called extremely amenable if every its continuous action on a compact space has a fixed point. It turns out that many non-locally compact groups, e.g. Aut([0,1],μ) or the unitary group of ℓ₂, share this property. I will discuss extreme amenability of abelian groups, which is connected with the study of characters of these groups. Given a submeasure μ on an algebra of subsets of a set X, the group L₀(μ) is the set of all real-valued measurable functions on X with the pointwise addition. L₀(μ) is endowed with the topology of convergence in submeasure μ.

Answering a question of Farah, Solecki and Pestov, and generalizing earlier results of Herer-Christensen, Glasner, Furstenberg-Weiss, Pestov, and Farah-Solecki, I will show that the group L_₀(μ) is extremely amenable if and only if μ is a diffused submeasure. This implies that a group of the form L₀(μ) is extremely amenable if and only if it has no nontrivial characters. The proof will be based on estimates of chromatic numbers of certain graphs on Zⁿ and a construction of a certain family of simplicial complexes.