McDuff superrigidity for group II$_1$ factors

TL;DR

This paper introduces McDuff superrigidity for certain ICC groups, showing that their von Neumann algebras uniquely determine the group structure up to a product with an amenable group, blending geometric group theory and operator algebras.

Contribution

It identifies the first examples of ICC groups with McDuff von Neumann algebras that exhibit a new superrigidity phenomenon, linking algebraic and geometric properties.

Findings

Existence of ICC groups with McDuff von Neumann algebras

Group von Neumann algebra determines the group up to a product with an amenable group

New rigidity phenomenon connecting group structure and von Neumann algebra properties

Abstract

Developing new techniques at the interface of geometric group theory and von Neumann algebras, we identify the first examples of ICC groups $G$ whose von Neumann algebras are McDuff and exhibit a new rigidity phenomenon, termed McDuff superrigidity: an arbitrary group $H$ satisfying $\mathcal{L}(H)\cong \mathcal{L}(G)$ decomposes as $H \cong G\times A$ for an ICC amenable group $A$.