Some $OE$ and $W^*$-rigidity results for actions by wreath product groups

TL;DR

This paper employs deformation-rigidity theory within von Neumann algebras to establish orbit equivalence and W*-rigidity results for actions of wreath product groups, revealing deep structural invariances.

Contribution

It identifies conditions under which wreath product groups exhibit orbit equivalence and W*-rigidity, advancing understanding of their von Neumann algebraic properties.

Findings

Wreath product groups with property (T) show orbit equivalence rigidity.

Measure equivalence of wreath products implies measure equivalence of their factors.

W*-rigidity results demonstrate von Neumann algebra invariance under certain group actions.

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$^*$-rigidity results) are also obtained.