Examples of W$^*$ and C$^*$-superrigid product groups

TL;DR

This paper introduces a new class of amalgamated free product groups, extending product rigidity results and providing examples of groups that are both W*- and C*-superrigid, advancing the understanding of group von Neumann algebras.

Contribution

It defines the class al_{AFP} of groups for which product rigidity holds, including known superrigid groups, and shows these groups are both W*- and C*-superrigid.

Findings

al_{AFP} includes W^* and C^*-superrigid groups.

Product rigidity results extend to this new class.

Examples of groups that are both W^* and C^*-superrigid are provided.

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 $L(G_1\times\dots\times G_n)\cong L(H)$, then there exists a product decomposition $H=H_1\times\dots\times H_n$ such that $L(H_i)$ is stably isomorphic to $L(G_i)$, for any $1\leq i\leq n$. The class $\mathcal C_{AFP}$ contains $W^*$ and $C^*$-superrigid groups from [CD-AD20]. Consequently, we obtain examples of product groups that are both $W^*$ and $C^*$-superrigid.