Conjugacy, orbit equivalence and classification of measure-preserving group actions

Abstract

We prove that if G is a countable discrete group with property (T) over an infinite subgroup H G which contains an infinite Abelian subgroup or is normal, then G has continuum many orbit inequivalent measure preserving a.e. free ergodic actions on a standard Borel probability space. Further, we obtain that the measure preserving a.e. free ergodic actions of such a G cannot be classified up to orbit equivalence be a reasonable assignment of countable structures as complete invariants. We also obtain a strengthening and a new proof of a nonclassification result of Foreman and Weiss for conjugacy of measure preserving ergodic, a.e. free actions of discrete countable groups.

Localized cohomology and some applications of Popa’s cocycle superrigidity theorem

“…Remark 2. The results of [21] imply that under fairly general conditions, if a countably infinite group Γ has the relative property (T), then both conjugacy and orbit equivalence of p.m.p. ergodic a.e.…”

“…free actions of Γ is not classifiable by "countable structures" (as defined in [10]), which in particular implies that it is not possible to Borel reduce conjugacy and orbit equivalence in this setting to ABEL ℵ0 . Thus we have the following: The normality condition in Corollary 5.6 can be replaced with the technically weaker notion of being index stable; we refer the reader to the last section of [21] for details.…”

“…Before stating the next Lemma, we recall various basic notions from [21]. Let E be a measure preserving equivalence relation.…”

“…A quick introduction to the significance of this notion is given in the introduction of [21]. Here it suffices to say that ≤ B gives a degree theory for the complexity of equivalence relations.…”

“…If f : X → Y is Borel, X, Y Polish spaces, and µ a measure on X, then we denote by f [µ] the push-forward measure on Y . (Note that our notation differs from [14] here, but is in line with [21]):…”