Topological dynamics of Polish group extensions

Abstract

We consider a short exact sequence $1\to H\to G\to K\to 1$ of Polish groups and consider what can be deduced about the dynamics of $G$ given information about the dynamics of $H$ and $K$. We prove that if the respective universal minimal flows $M(H)$ and $M(K)$ are metrizable, then so is $M(G)$. Furthermore, we show that if $M(H)$ and $M(K)$ are metrizable and both $H$ and $K$ are uniquely ergodic, then so is $G$. We then discuss several examples of these phenomena