New examples of W${}^{\ast }$ and C${}^{\ast }$-superrigid groups

Abstract

A group $G$ is called $W^{\ast }$-superrigid (resp. $C^{\ast }$-superrigid) if it is completely recognizable from its von Neumann algebra $L(G)$ (resp. reduced $C^{\ast }$-algebra $C_r^{\ast }(G)$). Developing new technical aspects in Popa's deformation/rigidity theory we introduce several new classes of $W^{\ast }$-superrigid groups which appear as direct products, semidirect products with non-amenable core and iterations of amalgamated free products and HNN-extensions. As a byproduct we obtain new rigidity results in $C^{\ast }$-algebra theory including additional examples of $C^{\ast }$-superrigid groups and explicit computations of symmetries of reduced group $C^{\ast }$-algebras.