Hyper-hyperfiniteness and complexity

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 relationsis as high as possible, namely, ${\Sigma }_2^1$-complete. We also establish an implication between the question of the effectivity of hyperfiniteness and its complexity.