On Polish groups admitting non-essentially countable actions

Abstract

It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.

Dynamical obstructions to classification by (co)homology and other TSI-group invariants

“…Similarly, Hjorth's turbulence theory [Hjo00] provides obstructions to classifiability by countable structures. Finally, storminess [Hjo05], as well as local approximability [KMPZ20], are both dynamical obstruction to being essentially countable.…”

“…In fact, in Section 2 we define a P -jump operator E → E [P ] , for every Polish permutation group P , which is a common generalization of the Friedman-Stanley jump and the jump operators defined in [CC]. It turns out that P -jumps are particularly natural in the context of the generalized Bernoulli shifts from [KMPZ20]: if E is the orbit equivalence relation of the generalized Bernoulli shift of the Polish permutation group Q, then E [P ] is the orbit equivalence relation of the generalized Bernoulli shift of (P Wr Q).…”

“…In this section, for every Polish permutation group P of a countable set we introduce a P -jump operator for G-spaces and equivalence relations, in the spirit of [CC]. We then draw some connections with the theory of generalized Bernoulli shifts from [KMPZ20]. We also prove Theorem 1.5, and derive Corollary 1.6 and Corollary 1.7.…”

“…This indeed generalizes the classical Bernoulli shift Γ R Γ for countable disrete groups, where (g, x) → g • x with (g • x)(γ) = x(g −1 γ). In [KMPZ20] it was shown that the Borel reduction complexity of the orbit equivalence relation of P R N is often a reflection of the dynamical properties of P . In addition to P above, consider a Polish permutation group Q on some set M .…”

Strong ergodicity phenomena for Bernoulli shifts of bounded algebraic dimension

“…When it comes to Polish permutation groups P ≤ Sym(N), an action which often serves as bridge between structural and dynamical properties of P is the Bernoulli shift P R N of P . For example, if E(P ) is its orbit equivalence relation, then: (1) E(P ) is smooth if and only if P is compact, see [KMPZ20];…”