New examples of W$^*$ and C$^*$-superrigid groups

TL;DR

This paper introduces new classes of groups that are uniquely determined by their associated operator algebras, expanding the understanding of superrigidity in von Neumann and C*-algebra contexts.

Contribution

It develops new techniques in deformation/rigidity theory to identify classes of W*-superrigid groups, including complex constructions like direct products and amalgamated free products.

Findings

New classes of W*-superrigid groups identified

Additional examples of C*-superrigid groups provided

Explicit symmetry computations of reduced group C*-algebras

Abstract

A group $G$ is called $W^*$-superrigid (resp. $C^*$-superrigid) if it is completely recognizable from its von Neumann algebra $L(G)$ (resp. reduced $C^*$-algebra $C_r^*(G)$). Developing new technical aspects in Popa's deformation/rigidity theory we introduce several new classes of $W^*$-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 $C^*$-algebra theory including additional examples of $C^*$-superrigid groups and explicit computations of symmetries of reduced group $C^*$-algebras.