Logic Seminar

Please note that the time is PST

We define the αα-balanced Polish groups for α<ω1α<ω1 and show that these form a stratification of the CLI Polish groups. Moreover, for every αα, we give an example of an αα-balanced Polish group and a continuous action of it on a Polish space such that the resulting orbit equivalence relation is not Borel-reducible to any such action of a ββ-balanced Polish group for any β<αβ<α. Recall that the CLI Polish groups are those that have compatible complete left-invariant metrics, where a metric dd is invariant if d(h,h′)=d(gh,gh′)d(h,h′)=d(gh,gh′) for every gg, hh, and h′h′. Our notion of αα-balanced is heavily inspired by a model-theoretic rank of Deissler, and another rank notion of Malicki. We also show that this class is in fact Π11Π11-complete, strengthening a result of Malicki.

This is joint work with Aristotelis Panagiotopoulos.