Ergodic treeable equivalence relations are richly ergodic

Abstract

We study the density of ergodic subrelations in the canonical Polish space of subequivalence relations of a probability-measure-preserving countable Borel equivalence relation. Our main result establishes rich ergodicity for ergodic treeable equivalence relations. The proof combines a strongly increasing exhaustion with relative ergodic hyperfinite decomposition. As an application, for every $2\leq r<\infty$, we show that the product of the dyadic odometer action of $\mathbb{Z}$ with the Bernoulli action of $\mathbb{F}_r$ has a richly ergodic orbit relation, whereas the Bernoulli action of $\mathbb{Z}\times\mathbb{F}_r$ does not. Thus rich ergodicity of a free ergodic action is not determined by the acting group.