Some $OE$ and $W^{\ast }$-rigidity results for actions by wreath product groups

Abstract

We use deformation-rigidity theory in von Neumann algebra framework to study probability measure preserving actions by wreath product groups. In particular, we single out large families of wreath products groups satisfying various type of orbit equivalence (OE) rigidity. For instance, we show that whenever $H$, $K$, $\G$, $\La$ are icc, property (T) groups such that $H\wr \G$ is measure equivalent to $K\wr \La$ then automatically $\G$ is measure equivalent to $\La$ and $H^{\G}$ is measure equivalent to $K^{\La}$. Rigidity results for von Neumann algebras arising from certain actions of such groups (i.e. W${}^{\ast }$-rigidity results) are also obtained.

28 pages

References in corpus (3)

• On a Class of Type II${}_1$ Factors with Betti Numbers Invariants
• Some unique group-measure space decomposition results
• Cocycle superrigidity for profinite actions of irreducible lattices

Cited by in corpus (2)

• A Unique Prime Decomposition Result for Wreath Product Factors
• Classification of tensor decompositions for II${}_1$ factors