The class and dynamics of α-balanced Polish groups

A Polish group is TSI if it admits a two-side invariant metric. It is CLI if it admits complete and left-invariant metric. The class of CLI groups contains every TSI group but there are many CLI groups that fail to be TSI. In this talk we will introduce the class of α-balanced Polish groups where α ranges over all countable ordinals. We will show that these classes completely stratify the space between TSI and CLI. We will also introduce "generic α-unbalancedness": a turbulence-like obstruction to classification by actions of α-balanced Polish groups. Finally, for each α we will provide an action of an α-balanced Polish group whose orbit equivalence relation is not classifiable by actions of any β-balanced Polish group with β<α. This is joint work with Shaun Allison.