An order analysis of hyperfinite Borel equivalence relations

TL;DR

This paper investigates the structure of hyperfinite Borel equivalence relations with Borel orderings, establishing a dichotomy for compatibility and linking self-compatibility of orderings to hyperfiniteness.

Contribution

It introduces a notion of compatibility for Borel $bZ$-orderings, proves a dichotomy theorem, and connects self-compatible orderings with hyperfiniteness of equivalence relations.

Findings

Incompatible pairs embed a canonical $E_0$-pair.

Compatible pairs are characterized by the dichotomy theorem.

Self-compatible $bZ^2$-orderings imply hyperfiniteness.

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.