A converse to Dye’s theorem

Hjorth, Greg

doi:10.1090/S0002-9947-04-03672-4

A converse to Dye’s theorem
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by Greg Hjorth

Trans. Amer. Math. Soc. 357 (2005), 3083-3103

DOI: https://doi.org/10.1090/S0002-9947-04-03672-4

Published electronically: July 22, 2004

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Abstract:

Every non-amenable countable group induces orbit inequivalent ergodic equivalence relations on standard Borel probability spaces. Not every free, ergodic, measure preserving action of $\mathbb {F}_2$ on a standard Borel probability space is orbit equivalent to an action of a countable group on an inverse limit of finite spaces. There is a treeable non-hyperfinite Borel equivalence relation which is not universal for treeable in the $\leq _B$ ordering.

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