W${}^{\ast }$-superrigidity for wreath products with groups having positive first ${\ell }^2$-Betti number

Abstract

In [BV12] we have proven that, for all hyperbolic groups and for all non-trivial free products $\Gamma$, the left-right wreath product group $G:=(Z/2Z{)}^{(\Gamma )}⋊(\Gamma \times \Gamma )$ is W${}^{\ast }$-superrigid. In this paper, we extend this result to other classes of countable groups. More precisely, we prove that for weakly amenable groups $\Gamma$ having positive first ${\ell }^2$-Betti number, the same wreath product $G$ is W${}^{\ast }$-superrigid.

v2: minor changes, to appear in Int. J. Math

References in corpus (2)

• Strong solidity of group factors from lattices in SO(n,1) and SU(n,1)
• W*-superrigidity for group von Neumann algebras of left-right wreath products