W*-superrigidity for group von Neumann algebras of left-right wreath products

TL;DR

This paper proves that for many nonamenable groups, the associated group von Neumann algebra uniquely determines the group, establishing W*-superrigidity for a class of wreath product groups.

Contribution

It establishes W*-superrigidity for left-right wreath products of certain nonamenable groups, including hyperbolic groups and free products, showing the von Neumann algebra fully encodes the group.

Findings

Group von Neumann algebra LG uniquely determines G for these groups.

W*-superrigidity holds for wreath products of hyperbolic groups and free products.

Any isomorphism of LG with another group von Neumann algebra implies the groups are isomorphic.

Abstract

We prove that for many nonamenable groups \Gamma, including all hyperbolic groups and all nontrivial free products, the left-right wreath product group G := (Z/2Z)^(\Gamma) \rtimes (\Gamma \times \Gamma) is W*-superrigid. This means that the group von Neumann algebra LG entirely remembers G. More precisely, if LG is isomorphic with L\Lambda for an arbitrary countable group \Lambda, then \Lambda must be isomorphic with G.