W$^*$-superrigidity for property (T) groups with infinite center

Abstract: We propose to study a natural version of Connes' Rigidity Conjecture that involves property (T) groups with infinite center. Utilizing techniques at the intersection of von Neumann algebras and geometric group theory, we establish several cases where this conjecture holds. In particular, we provide the first example of a W$^*$-superrigid property (T) group with infinite center. In the course of proving our main results, we also generalize the main W$^*$-superrigidity result from \cite{cios22} to twisted group factors.