Topological dynamics of automorphism groups, ultrafilter combinatorics, and the Generic Point Problem

Abstract:

For $G$ a closed subgroup of $S_{\infty }$, we provide a precise combinatorial characterization of when the universal minimal flow $M(G)$ is metrizable. In particular, each such instance fits into the framework of metrizable flows developed by Kechris, Pestov, and Todorčević and by Nguyen Van Thé; as a consequence, each $G$ with metrizable universal minimal flow has the generic point property, i.e. every minimal $G$-flow has a point whose orbit is comeager. This solves the Generic Point Problem raised by Angel, Kechris, and Lyons for closed subgroups of $S_\infty$.