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


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).


Ionuţ Chifan

The University of Iowa, Iowa City, IA, USA

Adrian Ioana

University of California San Diego, La Jolla, CA, USA

Denis Osin

Vanderbilt University, Nashville, TN, USA

Bin Sun

Mathematical Institute, University of Oxford, Oxford, UK