W*-superrigidity for wreath products with groups having positive first l2-Betti number

In [M. Berbec and S. Vaes, W*-superrigidity for group von Neumann algebras of leftright wreath products, Proc. London Math. Soc. 108 (2014) 1116 1152] we have proven that, for all hyperbolic groups and for all nontrivial free products Gamma, the left-right wreath product group G := (Z/2Z)((Gamma)) x (Gamma x Gamma) is W*-superrigid, in the sense that its group von Neumann algebra LG completely remembers the group G. In this paper, we extend this result to other classes of countable groups. More precisely, we prove that for weakly amenable groups G having positive first l(2)-Betti number, the same wreath product group G is W*-superrigid.