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

TL;DR

This paper introduces wreath-like groups, a new class related to wreath products, and proves that certain property (T) groups within this class are W*-superrigid, meaning their von Neumann algebras uniquely determine the group.

Contribution

It establishes W*-superrigidity for a new class of wreath-like groups with property (T), providing numerous non-isomorphic examples.

Findings

Wreath-like groups with property (T) are W*-superrigid.

First examples of W*-superrigid groups with property (T) are constructed.

Countably many non-isomorphic W*-superrigid groups are identified.

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).