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 $\mathrm{W}^\ast$-superrigid: the group von Neumann algebra $\mathrm{L}(G)$ remembers the isomorphism class of $G$. This allows us to provide the first examples (in fact, $2^{\aleph_0}$ pairwise non-isomorphic examples) of $\mathrm{W}^\ast$-superrigid groups with property (T).
Ann. of Math. (2)
198(3):
1261-1303
(November 2023).
DOI: 10.4007/annals.2023.198.3.6
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