Wreath-like products of groups and their von Neumann algebras I: $\mathrm{W}^\ast $-superrigidity

Abstract

We introduce a new class of groups called wreath-like products. These groups are close relatives of the classical wreath products and arise naturally in the context of group theoretic Dehn filling. Unlike ordinary wreath products, many wreath-like products have Kazhdan’s property (T). In this paper, we prove that any group $G$ in a natural family of wreath-like products with property (T) is W$^*$-superrigid: the group von Neumann algebra $\text{L}(G)$ remembers the isomorphism class of $G$. This allows us to provide the first examples (in fact, $2^{\aleph _0}$ pairwise non-isomorphic examples) of W$^*$-superrigid groups with property (T).