Borel equivalence relations induced by actions of tsi Polish groups

TL;DR

This paper investigates Borel equivalence relations arising from actions of tsi Polish groups, providing a characterization for when they can be classified by countable structures and identifying a base class for non-classification.

Contribution

It introduces a new characterization for classification by countable structures for these equivalence relations and identifies a base class for non-classification under Borel reducibility.

Findings

Characterization of when such equivalence relations admit classification by countable structures.

Identification of a base class for non-classification by countable structures.

Proof that certain equivalence relations Borel reduce to others under specific conditions.

Abstract

We study Borel equivalence relations induced by Borel actions of tsi Polish groups on standard Borel spaces. We characterize when such an equivalence relation admits classification by countable structures using a variant of the $\mathbb G_0$-dichotomy. In particular, we find a class that serves as a base for non-classification by countable structures for these equivalence relations under Borel reducibility. We use this characterization together with the result of [B. D. Miller, to appear in the Journal of Mathematical Logic] to show that if such an equivalence relation admits classification by countable structures but it is not essentially countable, then the equivalence relation ${\mathbb E}^{\mathbb N}_0=\mathbb E_3$ Borel reduces to it.