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

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.