Examples of W${}^{\ast }$ and C${}^{\ast }$-superrigid product groups

Abstract

We provide a new large class ${\mathcal{C}}_{AFP}$ of amalgamated free product groups for which the product rigidity result from [CdSS15] holds: if $G_1,\dots ,G_n\in {\mathcal{C}}_{AFP}$ and $H$ is any group such that $L(G_1\times \cdots \times G_n)\cong L(H)$, then there exists a product decomposition $H=H_1\times \cdots \times H_n$ such that $L(H_i)$ is stably isomorphic to $L(G_i)$, for any $1\le i\le n$. The class ${\mathcal{C}}_{AFP}$ contains $W^{\ast }$ and $C^{\ast }$-superrigid groups from [CD-AD20]. Consequently, we obtain examples of product groups that are both $W^{\ast }$ and $C^{\ast }$-superrigid.