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

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.

💡 Research Summary

The paper introduces a broad new class of groups, denoted 𝒞₍AFP₎, consisting of amalgamated free products G = G₁ ∗_Σ G₂ where Σ is a common amenable icc subgroup of the two factors and each factor splits as a product of two icc, non‑amenable, relatively solid groups. When additional technical conditions (iii) and (iv) are satisfied, the subclass 𝒞₀₍AFP₎ is obtained; this subclass already contains the W∗‑ and C∗‑superrigid groups constructed in the authors’ earlier work