Actions of tame abelian product groups

TL;DR

This paper determines the precise potential complexity bound for orbit equivalence relations of tame abelian product groups, improving previous bounds and establishing optimality through a combination of construction and set-theoretic analysis.

Contribution

It establishes that the optimal potential complexity bound for actions of tame abelian product groups is $D(\mathbf{\Pi}^0_5)$, refining earlier bounds and demonstrating optimality.

Findings

The potential complexity bound is $D(\mathbf{\Pi}^0_5)$ for such groups.

Constructs an action not potentially $\mathbf{\Pi}^0_5$, proving the bound's sharpness.

Uses forcing over models where choice fails to analyze lower bounds.

Abstract

A Polish group $G$ is tame if for any continuous action of $G$, the corresponding orbit equivalence relation is Borel. When $G = \prod_n \Gamma_n$ for countable abelian $\Gamma_n$, Solecki (1995) gave a characterization for when $G$ is tame. Ding and Gao (2017) showed that for such $G$, the orbit equivalence relation must in fact be potentially $\mathbf{\Pi}^0_6$, while conjecturing that the optimal bound could be $\mathbf{\Pi}^0_3$. We show that the optimal bound is $D(\mathbf{\Pi}^0_5)$ by constructing an action of such a group $G$ which is not potentially $\mathbf{\Pi}^0_5$, and show how to modify the analysis of Ding and Gao to get this slightly better upper bound. It follows, using the results of Hjorth, Kechris, and Louvaeu (1998), that this is the optimal bound for the potential complexity of actions of tame abelian product groups. Our lower-bound analysis involves forcing over models of set theory where choice fails for sequences of finite sets.