Dynamical obstructions for classification by actions of TSI groups

Shaun Allison, Aristotelis Panagiotopoulos

Published 2020-04-16Version 1

We introduce a dynamical condition for Polish $G$-spaces, in the spirit of Hjorth's turbulence theory, which implies non-classifiability by actions of Polish TSI-groups. These are the groups which admit a metric that is invariant from both sides. We show that the wreath product of any two non-compact subgroups of $S_{\infty}$ admits an action whose orbit equivalence relation is generically ergodic against actions of TSI-groups, and deduce that there is an orbit equivalence relation of a CLI group which is not classifiable by TSI group actions. Finally, we show that Morita equivalence of continuous-trace $C^*$-algebras, as well as isomorphism of Hermitian line bundles, are not classifiable by TSI-group actions.