Ergodic treeable equivalence relations are richly ergodic

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≤ r<∞, we show that the product of the dyadic odometer action of Z with the Bernoulli action of Fᵣ has a richly ergodic orbit relation, whereas the Bernoulli action of Zᵣ does not. Thus rich ergodicity of a free ergodic action is not determined by the acting group.