We show that the automorphism group Aut([0,1],λ) of the Lebesgue measure has no non-trivial subgroups of index $<2^{\omega }$.

We investigate the connections between Ramsey properties of Fraïssé classes and the universal minimal flow $M(G_{)}$ of the automorphism group $G$ of their Fraïssé limits. As an extension of a result of Kechris, Pestov and Todorcevic (2005) we show that if the class has finite Ramsey degree for embeddings, then this degree equals the size of $M(G_{)}$. We give a partial answer to a question of Angel, Kechris and Lyons (2014) showing that if is a relational Ramsey class and $G$ is amenable, then $M(G_{)}$ admits a unique invariant...

We develop methods for studying transition operators on metric spaces that are invariant under a co-compact group which acts properly. A basic requirement is a decomposition of such operators with respect to the group orbits. We then introduce reduced transition operators on the compact factor space whose norms and spectral radii are upper bounds for the Lp-norms and spectral radii of the original operator. If the group is amenable then the spectral radii of the original and reduced operators coincide,...