Extreme amenability of L0, a Ramsey theorem, and Lévy groups

We show that L0(ϕ,H) is extremely amenable for any diffused submeasure ϕ and any solvable compact group H. This extends results of Herer–Christensen, and of Glasner and Furstenberg–Weiss. Proofs of these earlier results used spectral theory or concentration of measure. Our argument is based on a new Ramsey theorem proved using ideas coming from combinatorial applications of algebraic topological methods. Using this work, we give an example of a group which is extremely amenable and contains an increasing sequence of compact subgroups with dense union, but which does not contain a Lévy sequence of compact subgroups with dense union. This answers a question of Pestov. We also show that many Lévy groups have non-Lévy sequences, answering another question of Pestov.