TOPOLOGICAL DYNAMICS OF AUTOMORPHISM GROUPS, ULTRAFILTER COMBINATORICS, AND THE GENERIC POINT PROBLEM

摘要

For G a closed subgroup of S-infinity, 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 Todorcevic and by Nguyen Van The; 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-infinity.