I will survey some recent results on rigidity and automorphisms of von Neumann algebras of groups with Kazhdan property (T) obtained in a series of joint papers with I. Chifan, A. Ioana, and B. Sun. Specifically, we show that certain groups, constructed via a group-theoretic version of Dehn filling in 3-manifolds, satisfy several conjectures proposed by A. Connes, V. Jones, and S. Popa. Previously, no nontrivial examples of groups satisfying these conjectures were known. At the core of our approach is the new notion of a wreath-like product of groups, which seems to be of independent interest.
