Group measure space decomposition of II1 factors and W*-superrigidity

We prove a “unique crossed product decomposition” result for group measure space II₁ factors L ^∞(X)⋊Γ arising from arbitrary free ergodic probability measure preserving (p.m.p.) actions of groups Γ in a fairly large family G\mathcal{G}, which contains all free products of a Kazhdan group and a non-trivial group, as well as certain amalgamated free products over an amenable subgroup. We deduce that if T _n denotes the group of upper triangular matrices in PSL (n,ℤ), then any free, mixing p.m.p. action of G = \operatornamePSL(n,\mathbbZ)*_Tn\operatornamePSL(n,\mathbbZ)\Gamma=\operatorname{PSL}(n,\mathbb{Z})*_{T_{n}}\operatorname{PSL}(n,\mathbb{Z}) is W^∗-superrigid, i.e. any isomorphism between L ^∞(X)⋊Γ and an arbitrary group measure space factor L ^∞(Y)⋊Λ, comes from a conjugacy of the actions. We also prove that for many groups Γ in the family G\mathcal{G}, the Bernoulli actions of Γ are W^∗-superrigid.