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

Abstract

We use deformation-rigidity theory in the von Neumann algebra framework to study probability measure preserving actions by wreath product groups. In particular, we single out large families of wreath product groups satisfying various types of orbit equivalence (OE) rigidity. For instance, we show that whenever H , K, Γ , Λ are icc, property (T) groups such that H Γ and K Λ admit stably orbit equivalent action σ and ρ such that σ | Γ , ρ| Λ , σ | H Γ , and ρ| K Λ are ergodic, then automatically σ Γ is stably orbit equivalent to ρ Λ and σ | H Γ is stably orbit equivalent to ρ| K Λ . Rigidity results for von Neumann algebras arising from certain actions of such groups (i.e. W * -rigidity results) are also obtained. Published by Elsevier Inc.

Measure equivalence rigidity via s-malleable deformations

“…Popa's pioneering work [Pop06b, Pop06c] allowed one to distinguish between the group von Neumann algebras of , as is an infinite property (T) group, while Ioana, Popa, and Vaes used a wreath product construction to obtain the first class of groups that are entirely remembered by their von Neumann algebras [IPV13]. Subsequently, several other rigidity results have been obtained for von Neumann algebras of wreath products including primeness, relative solidity, and product rigidity, see [Ioa07, Pop08, CI10, Ioa11, IPV13, SW13, CPS12, BV14, IM19, Dri21, CDD21]. Theorem A establishes a new general rigidity result for wreath product groups by showing that products of arbitrary non-amenable wreath product groups with amenable base satisfy an analogue of Monod and Shalom's unique prime factorization result.…”