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

Published 27 Mar 2014 in math.OA and math.GR | (1403.7110v2)

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 $G:=(Z/2Z)^{(\Gamma)} \rtimes (\Gamma \times \Gamma)$ is W$^{*$-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$^{*$-superrigid.}