Extreme amenability of \(L_0\), a Ramsey theorem, and Lévy groups (Q935061)

A topological group is called extremely amenable if each of its continuous actions on a compact space has a fixed point. In this article, the authors show that \(L_0(\varphi,H)\) is extremely amenable for any diffused submeasure \(\varphi\) and any solvable compact group \(H\). This extends results of Herer-Christensen, and of Glasner and Furstenberg-Weiss. The argument in the proof is based on a new Ramsey theorem proved using ideas coming from combinatorial applications of algebraic topological methods. Using this work, the authors 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 \textit{V. Pestov} [Dynamics of Infinite-dimensional Groups: The Ramsey-Dvoretzky-Milman phenomenon, Univ. Lect. Ser., Vol. 40, Am. Math. Soc., Providence, RI (2006; Zbl 1123.37003)]. The authors also show that many Lévy groups have non-Lévy sequences, what answers another question of Pestov.