Connes' rigidity conjecture for lattices in higher-rank and other property (T) local-field groups (W*-superrigidity of lattice group factors)

Principal (Theorem 1.2): for $\Gamma$ in the class $\mathscr K$ (ICC groups commensurable with lattices in finite products of noncompact property (T) groups $\mathbf H_i(k_i)^+$ over $\mathbb R$, $\mathbb C$ or $p$-adic fields), any ICC $\Lambda$ and arbitrary scalar 2-cocycles, every bifinite $L_\mu(\Gamma)$-$L_\omega(\Lambda)$ correspondence is a summand of finite sums of models built from finite-index subgroup isomorphisms and finite-dimensional projective representations. Companion: $L_\omega(\Lambda)^t\cong L_\nu(\Pi)^s$ for $\Lambda\in\mathscr K$ and ICC $\Pi$ iff $t=s$ and the groups are isomorphic with cohomologous cocycles, every such isomorphism being implemented canonically; these factors have trivial fundamental group (e.g. $L(\mathrm{SL}_3(\mathbb Z))$). Not shown: lattices in rank-one groups without property (T), or general property (T) groups.

Principal (Theorem 1.2): for $\Gamma$ in the class $\mathscr K$ (ICC groups commensurable with lattices in finite products of noncompact property (T) groups $\mathbf H_i(k_i)^+$ over $\mathbb R$, $\mathbb C$ or $p$-adic fields), any ICC $\Lambda$ and arbitrary scalar 2-cocycles, every bifinite $L_\mu(\Gamma)$-$L_\omega(\Lambda)$ correspondence is a summand of finite sums of models built from finite-index subgroup isomorphisms and finite-dimensional projective representations. Companion: $L_\omega(\Lambda)^t\cong L_\nu(\Pi)^s$ for $\Lambda\in\mathscr K$ and ICC $\Pi$ iff $t=s$ and the groups are isomorphic with cohomologous cocycles, every such isomorphism being implemented canonically; these factors have trivial fundamental group (e.g. $L(\mathrm{SL}_3(\mathbb Z))$). Not shown: lattices in rank-one groups without property (T), or general property (T) groups.