The class and dynamics of α-balanced Polish groups

For each ordinal α<ω₁, we introduce the class of α-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 α-balancedness is an initial segment of a regular coanalytic rank. In the spirit of Hjorth's turbulence theory we also introduce "generic α-unbalancedness": a new dynamical condition for Polish G-spaces which serves as an obstruction to classification by actions of α-balanced Polish groups. We use this to provide, for each α<ω₁, an action of an α-balanced Polish group whose orbit equivalence relation is strongly generically ergodic against actions of any β-balanced Polish group with β<α.