Hyper-hyperfiniteness and complexity

TL;DR

This paper explores the complexity of hyperfinite countable Borel equivalence relations, demonstrating that if a certain higher-level relation exists, then their complexity reaches the maximum known level, ^1_2-complete.

Contribution

It establishes that the complexity of hyperfinite countable Borel equivalence relations can be as high as ^1_2-complete under specific conditions.

Findings

Hyper-hyperfinite relations imply maximum complexity for hyperfinite relations.

The complexity of hyperfinite relations can reach ^1_2-complete.

Conditional existence of higher-level relations affects the classification of hyperfinite relations.

Abstract

We show that if there exists a countable Borel equivalence relation which is hyper-hyperfinite but not hyperfinite then the complexity of hyperfinite countable Borel equivalence relations is as high as possible, namely, $\Sigma^1_2$-complete.