Projectivity in topological dynamics
Jashan Bal
TL;DR
This work extends the theory of projectivity to topological dynamics for Polish groups by studying $G$-flows and affine $G$-flows. It introduces proximally irreducible extensions and uses them to characterize extreme amenability, strong amenability, and amenability of closed subgroups via the Samuel compactification $ ext{Sa}(G/H)$ and associated affine flows like $P( ext{Sa}(G/H))$. The paper proves that $H$ is extremely amenable iff $ ext{Sa}(G/H)$ is $G$-projective, and $H$ is amenable iff $P( ext{Sa}(G/H))$ is affine $G$-projective, connecting dynamical irreducibility with operator-algebraic injectivity. It also develops affine $G$-projective covers, Gleason-type completions, and a structure theorem for the universal minimal proximal flow $ ext{Pi}(G)$, addressing Zucker’s open question on when $ ext{Pi}(G)$ is metrizable or has a comeager orbit. Collectively, these results illuminate the landscape of dynamical irreducibility and amenability through a unified framework of (affine) $G$-flows and their Samuel boundaries, with implications for structural Ramsey theory and the theory of universal dynamical objects.
Abstract
We study projectivity in the category of $G$-flows and affine $G$-flows for Polish groups $G$. We also introduce the notion of \emph{proximally irreducible} extensions between affine $G$-flows. Using this we provide a characterization of extreme amenability, strong amenability, and amenability for closed subgroups $H \leq G$ in terms of certain ``dynamical irreducibility'' properties of the Samuel compactification of $G/H$. We then apply this to answer an open question of Zucker by proving a structure theorem for when the universal minimal proximal flow of $G$ is metrizable or contains a comeager orbit.