An order analysis of hyperfinite Borel equivalence relations

TL;DR

The paper advances the understanding of hyperfinite Borel equivalence relations by introducing a compatibility framework for two Borel $\mathbb{Z}$-orderings and proving a dichotomy: either the orderings are compatible or a canonical obstruction arising from $E_0$ embeds in a way that preserves the order structure. An effective version of this dichotomy is established via $\Delta^1_1$ tools, yielding a clear alternative between monotone subsets and $E_0$-reductions. Extending the framework, the authors show that hyperfinite-over-hyperfinite relations with a self-compatible $\mathbb{Z}^2$-ordering are in fact hyperfinite, using a combination of forcing and DJK-type arguments. Together, these results contribute a unified perspective on hyperfinite, hyper-hyperfinite, and hyperfinite-over-hyperfinite classes and relate to broader questions about equivalence relations under Borel reducibility.

Abstract

In this paper we first consider hyperfinite Borel equivalence relations with a pair of Borel $\mathbb{Z}$-orderings. We define a notion of compatibility between such pairs, and prove a dichotomy theorem which characterizes exactly when a pair of Borel $\mathbb{Z}$-orderings are compatible with each other. We show that, if a pair of Borel $\mathbb{Z}$-orderings are incompatible, then a canonical incompatible pair of Borel $\mathbb{Z}$-orderings of $E_0$ can be Borel embedded into the given pair. We then consider hyperfinite-over-finite equivalence relations, which are countable Borel equivalence relations admitting Borel $\mathbb{Z}^2$-orderings. We show that if a hyperfinite-over-hyperfinite equivalence relation $E$ admits a Borel $\mathbb{Z}^2$-ordering which is self-compatible, then $E$ is hyperfinite.

Theorems & Definitions (46)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Definition 2.4
• Theorem 2.5
• Definition 2.6
• Proposition 2.7
• proof