Topological groups with tractable minimal dynamics

TL;DR

This paper introduces the class of topological groups with tractable minimal dynamics, generalizing previous classes, and develops new set-theoretic and dynamical tools to analyze their properties and connections to model theory.

Contribution

It defines the class TMD of topological groups with tractable minimal dynamics, generalizes results from GPP, and applies set-theoretic methods to study their structure and connections to model theory.

Findings

Characterizes TMD via an abstract KPT correspondence.

Shows TMD is _1 in the Levy hierarchy.

Generalizes Glasner's structure theorem and proves the revised Newelski conjecture.

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.