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

Abstract

For each ordinal $\alpha <{\omega }_1$, we introduce the class of $\alpha$-balanced Polish groups. These classes form a hierarchy which 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 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 $\beta <\alpha$.