Metrizable universal minimal flows of Polish groups have a comeagre orbit

Abstract

We prove that, whenever $G$ is a Polish group with metrizable universal minimal flow $M(G)$, there exists a comeagre orbit in $M(G)$. It then follows that there exists an extremely amenable, closed, coprecompact $G^{\ast }$ of $G$ such that $M(G)=\widehat{G/G^{\ast }}$.