To any countable group G, one associates its group von Neumann
algebra L(G), which is the closure of the group algebra in a weak
topology. A central question is: "How does L(G) depend on G?" In the
extreme case where L(G) completely remembers G, one says that G is
W*-superrigid. Several classes of W*-superrigid groups are known, but
they are all icc (infinite conjugacy classes). In particular, they are
far from having centre. In fact, commutativity often poses an
obstruction for rigidity. Nevertheless, in a recent work with Stefaan
Vaes we were able to find the first W*-superrigid groups which have
infinite centre. I will explain this result and how it relates to a
previous work on W*-superrigidity for cocycle twisted group von Neumann
algebras.