The class and dynamics of $\alpha$-balanced Polish groups

TL;DR

This paper introduces and explores the hierarchy of $oldsymbol{ extalpha}$-balanced Polish groups, connecting their properties to model theory, turbulence, and classification obstructions.

Contribution

It defines the class of $oldsymbol{ extalpha}$-balanced Polish groups, establishes their properties, and introduces new dynamical obstructions to classification.

Findings

$oldsymbol{ extalpha}$-balancedness forms a hierarchy between TSI and CLI groups.

Develops a boundedness principle for CLI groups using a coanalytic rank.

Introduces 'generic $oldsymbol{ extalpha}$-unbalancedness' as an obstruction to classification.

Abstract

For each ordinal $\alpha<\omega_1$, we introduce the class of $\alpha$-balanced Polish groups. These classes form a hierarchy that completely stratifies the space between the class of Polish groups admitting a two-side-invariant metric (TSI) and the class of Polish groups admitting a complete left-invariant metric (CLI). We establish various closure properties, provide connections to model theory, and we develop a boundedness principle for CLI groups by showing that $\alpha$-balancedness is an initial segment of a regular coanalytic rank. In the spirit of Hjorth's turbulence theory we also introduce "generic $\alpha$-unbalancedness": a new dynamical condition for Polish $G$-spaces which serves as an obstruction to classification by actions of $\alpha$-balanced Polish groups. We use this to provide, for each $\alpha<\omega_1$, an action of an $\alpha$-balanced Polish group whose orbit equivalence relation is strongly generically ergodic against actions of any $\beta$-balanced Polish group with $\beta<\alpha$.