Abstract
In the spirit of Hjorth’s turbulence theory, we introduce “unbalancedness”: a new dynamical obstruction to classifying orbit equivalence relations by actions of Polish groups which admit a two-sided invariant metric (TSI). Since abelian groups are TSI, unbalancedness can be used for identifying which classification problems cannot be solved by classical homology and cohomology theories. In terms of applications, we show that Morita equivalence of continuous-trace C ∗ C^* -algebras, as well as isomorphism of Hermitian line bundles, are not classifiable by actions of TSI groups. In the process, we show that the Wreath product of any two non-compact subgroups of S ∞ S_{\infty } admits an action whose orbit equivalence relation is generically ergodic against any action of a TSI group and we deduce that there is an orbit equivalence relation of a CLI group which is not classifiable by actions of TSI groups.
Highlights
The main goal of this paper is to provide an answer to the above problem for the case where C is the class of all Polish groups which admit a two side invariant (TSI) metric
Corollary 1.6 gives many examples of classification problems which are classifiable by countable structures but not by actions of two side invariant metric (TSI) groups
We show that coordinate free isomorphism between Hermitian line bundles and Morita equivalence between continuous-trace C∗-algebras are not classifiable by TSI-group actions
Summary
Storminess [Hjo05], as well as local approximability [KMPZ20], are both dynamical obstruction to being essentially countable These dynamical obstructons can be seen to answer the following general problem that was considered in [LP18]: if C is a class of Polish groups, we say that (X, E) is classifiable by C-group actions if it is Borel reducible to an orbit equivalence relation (Y, EYH ), where H is a group from C. If the Polish G-space X is generically unbalanced, the orbit equivalence relation EXG is not classifiable by TSI-group actions. Of the Bernoulli shift of (P WrN Q), has meager orbits and is generically ergodic with respect to actions of TSI Polish groups. Corollary 1.6 gives many examples of classification problems which are classifiable by countable structures but not by actions of TSI groups. The rest follows from Theorem 1.5 and the fact that the wreath product of CLI groups is CLI
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