W*-superrigidity for groups with infinite center

TL;DR

This paper constructs specific groups with infinite center that are uniquely determined by their von Neumann algebras, advancing the understanding of operator algebra rigidity.

Contribution

It introduces new examples of groups with infinite center that are W*-superrigid, combining quotient and cohomology rigidity techniques.

Findings

Groups with infinite center can be W*-superrigid

Von Neumann algebra fully encodes the group structure

Rigidity holds up to isomorphisms and virtual isomorphisms

Abstract

We construct discrete groups $G$ with infinite center that are nevertheless W*-superrigid, meaning that the group von Neumann algebra $L(G)$ fully remembers the group $G$. We obtain these rigidity results both up to isomorphisms and up to virtual isomorphisms of the groups and their von Neumann algebras. Our methods combine rigidity results for the quotient of these groups by their center with rigidity results for their 2-cohomology.