Borel reducibility and symmetric models

TL;DR

This paper establishes a new connection between Borel equivalence relations and symmetric models, proving conjectures and analyzing the complexity of certain equivalence relations using techniques from set theory and logic.

Contribution

It introduces a correspondence between Borel equivalence relations from group actions and symmetric models, solving longstanding conjectures and extending the analysis of equivalence relations beyond previous limits.

Findings

entity relation ifferentiates between Borel reducibility levels.

entity relation ifferentiates invariants for different group actions.

Constructs new equivalence relations with specific reducibility properties.

Abstract

We develop a correspondence between the study of Borel equivalence relations induced by closed subgroups of $S_\infty$, and the study of symmetric models and weak choice principles, and apply it to prove a conjecture of Hjorth-Kechris-Louveau (1998). For example, we show that the equivalence relation $\cong^\ast_{\omega+1,0}$ is strictly below $\cong^\ast_{\omega+1,<\omega}$ in Borel reducibility. By results of Hjorth-Kechris-Louveau, $\cong^\ast_{\omega+1,<\omega}$ provides invariants for $\Sigma^0_{\omega+1}$ equivalence relations induced by actions of $S_\infty$, while $\cong^\ast_{\omega+1,0}$ provides invariants for $\Sigma^0_{\omega+1}$ equivalence relations induced by actions of abelian closed subgroups of $S_\infty$. We further apply these techniques to study the Friedman-Stanley jumps. For example, we find an equivalence relation $F$, Borel bireducible with $=^{++}$, so that $F\restriction C$ is not Borel reducible to $=^{+}$ for any non-meager set $C$. This answers a question of Zapletal, arising from the results of Kanovei-Sabok-Zapletal (2013). For these proofs we analyze the symmetric models $M_n$, $n<\omega$, developed by Monro (1973), and extend the construction past $\omega$, through all countable ordinals. This answers a question of Karagila (2019).