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

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

2011-10-10

arxiv.org/abs/1110.2151v1

Mathematics

Operator Algebras

28 pages

Scientific paper

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.

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