Topological groups with tractable minimal dynamics

Abstract

A Polish group $G$ has the generic point property if any minimal $G$-flow admits a comeager orbit, or equivalently if the universal minimal flow (UMF) does. The class $\mathsf{GPP}$ of such Polish groups is a proper extension of the class $\sf{PCMD}$ of Polish groups with metrizable UMF. Motivated by analogous results for $\mathsf{PCMD}$, we define and explore a robust generalization of $\sf{GPP}$ which makes sense for all topological groups, thus defining the class $\mathsf{TMD}$ of topological groups with tractable minimal dynamics. These characterizations yield novel results even for $\mathsf{GPP}$; for instance, a Polish group is in $\mathsf{GPP}$ iff its UMF has no points of first countability. Motivated by work of Kechris, Pestov, and Todor\v{c}evi\'c that connects topological dynamics and structural Ramsey theory, we state and prove an abstract KPT correspondence which characterizes the class $\mathsf{TMD}$ and shows that $\mathsf{TMD}$ is $\Delta_1$ in the L\'evy hierarchy. We then develop set-theoretic methods which allow us to apply forcing and absoluteness arguments to generalize numerous results about $\mathsf{GPP}$ to all of $\mathsf{TMD}$. We also apply these new set-theoretic methods to first generalize parts of Glasner's structure theorem for minimal, metrizable tame flows to the non-metrizable setting, and then to prove the revised Newelski conjecture regarding definable NIP groups. We conclude by discussing some tantalizing connections between definable NIP groups and $\mathsf{TMD}$ groups.