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arXiv · 2610.40237

Ergodic treeable equivalence relations are richly ergodic

Wei Dai
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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.

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Wei Dai. 2026-10-29. Ergodic treeable equivalence relations are richly ergodic. https://arxiv.org/abs/2610.40237

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