Wreath-like products of groups and their von Neumann algebras I: $W^{\ast }$-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${}^{\ast }$-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${}^{\ast }$-superrigid groups with property (T).

The original paper (v1) has been split into three papers; results are strengthened and proofs are simplified. This is the first paper in the series, which contains results on W*-superrigidity. The second and third papers will focus on outer automorphisms and embeddings of von Neumann algebras of wreath-like products, respectively. To appear in the Annals of Mathematics

References in corpus (3)

• Classification of type III Bernoulli crossed products
• Noncommutative ergodic theory of higher rank lattices
• Type II${}_1$ factors with arbitrary countable endomorphism group