Topological Dynamics of Closed Subgroups of $S_\infty$

Abstract

For $G$ a closed subgroup of $S_{\infty}$, we provide an explicit characterization of the greatest $G$-ambit. Using this, we provide a precise characterization of when $G$ has metrizable universal minimal flow. In particular, each such instance fits into the framework of metrizable flows developed by Kechris-Pestov-Todorcevic and Nguyen Van Th\'e; as a consequence, each $G$ with metrizable universal minimal flow has the generic point property.