Ergodic treeable equivalence relations are richly ergodic

@misc{dai2026ergodic,
  title = {{Ergodic treeable equivalence relations are richly ergodic}},
  author = {Wei Dai},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40237},
  eprint = {2610.40237},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40237},
}

TY  - UNPB
TI  - Ergodic treeable equivalence relations are richly ergodic
AU  - Wei Dai
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.40237
UR  - https://arxiv.org/abs/2610.40237
AB  - 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 𝔽_r has a richly ergodic orbit relation, whereas the Bernoulli action of ℤ×𝔽_r does not. Thus rich ergodicity of a free ergodic action is not determined by the acting group.
ER  -