Measure equivalence rigidity and bi-exactness of groups

Abstract

We get three types of results on measurable group theory; direct product groups of Ozawa's class S groups, wreath product groups and amalgamated free products. We prove measure equivalence factorization results on direct product groups of Ozawa's class S groups. As consequences, Monod-Shalom type orbit equivalence rigidity theorems follow. We prove that if two wreath product groups A G, B Γ of nonamenable exact direct product groups G, Γ with amenable bases A, B are measure equivalent, then G and Γ are measure equivalent. We get Bass-Serre rigidity results on amalgamated free products of nonamenable exact direct product groups.

Some OE- and W⁎-rigidity results for actions by wreath product groups

“…This has been confirmed at the OE-level by Hiroki Sako in [34], who was able to prove that for a large class of groups Γ , the OE-class of an action H Γ X is completely determined by the OE-class of its restriction Γ X. More precisely, he showed that, if two actions by wreath products groups are orbit equivalent, H Γ ∼ =OE K Λ, where H , K are amenable and Γ , Λ are products of non-amenable, exact groups, then Γ ∼ =OE Λ.…”

“…Sako showed in [34] that a measure equivalence between two wreath products groups H Γ and K Λ, where H, K are amenable and Γ, Λ are products of non-amenable exact groups, implies the measure equivalence of the malnormal subgroups Γ and Λ. Further he showed that, given two stably orbit equivalent actions, σ and ρ, of such groups with σ | Γ and ρ| Λ ergodic, one has σ | Γ and ρ| Λ are stably orbit equivalent.…”

Rigidity for Von Neumann Algebras Given by Locally Compact Groups and Their Crossed Products

“…This result is similar to the cocycle superrigidity theorem of [MS04], where the target group is assumed to be a closed subgroup of the isometry group of a negatively curved space. We then deduce that Sako's orbit equivalence rigidity theorem [Sa09] for irreducible pmp actions G 1 × G 2 (X, µ) of nonamenable groups in class S stays valid in the locally compact setting. Recall here that a nonsingular action G 1 × G 2 (X, µ) of a direct product group is called irreducible if both G 1 and G 2 act ergodically.…”

Orbit Equivalence and Measured Group Theory

“…indecomposable as a non-trivial free product) (compare [IPP08,CH10]). See also [Sak09b] for similar results for some free products with amalgamation over amenable groups. Examples of MFI groups are provided by non-amenable groups with β (2) 1 = 0.…”