2024/12/07 by Gianluca Basso, Basso, Gianluca, Andy Zucker +1 · 2 citations
Computer Science · Mathematics · #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Logic (math.LO) #Topological and Geometric Data Analysis #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2412.05659
openalex publication_date 2024/12/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 GPP of such Polish groups is a proper extension of the class \sfPCMD of Polish groups with metrizable UMF. Motivated by analogous results for PCMD, we define and explore a robust generalization of \sfGPP which makes sense for all topological groups, thus defining the class TMD of topological groups with tractable minimal dynamics. These characterizations yield novel results even for GPP; for instance, a Polish group is in GPP iff its UMF has no points of first countability. Motivated by work of Kechris, Pestov, and Todorčević that connects topological dynamics and structural Ramsey theory, we state and prove an abstract KPT correspondence which characterizes the class TMD and shows that TMD is Δ1 in the Lévy hierarchy. We then develop set-theoretic methods which allow us to apply forcing and absoluteness arguments to generalize numerous results about GPP to all of 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 TMD groups.