Invariant subalgebras of von Neumann algebras arising from negatively curved groups

TL;DR

This paper proves that for a broad class of negatively curved groups, their von Neumann algebras have a unique invariant subalgebra structure, answering a key open question in operator algebra theory.

Contribution

It establishes the ISR property for von Neumann algebras of many negatively curved groups, combining geometric group theory and von Neumann algebra techniques, and extends results to groups with nontrivial (quasi)cohomology.

Findings

Proves ISR property for von Neumann algebras of acylindrically hyperbolic groups.

Shows the property holds for groups with positive first $L^2$-Betti number containing an infinite amenable subgroup.

Answers an open question by Amrutam and Jiang regarding invariant subalgebras.

Abstract

Using an interplay between geometric methods in group theory and soft von Neuman algebraic techniques we prove that for any icc, acylindrically hyperbolic group $\Gamma$ its von Neumann algebra $L(\Gamma)$ satisfies the so-called ISR property: \emph{any von Neumann subalgebra $N\subseteq L(\Gamma)$ that is normalized by all group elements in $\Gamma$ is of the form $N= L(\Sigma)$ for a normal subgroup $\Sigma \lhd \Gamma$.} In particular, this applies to all groups $\Gamma$ in each of the following classes: all icc (relatively) hyperbolic groups, most mapping class groups of surfaces, all outer automorphisms of free groups with at least three generators, most graph product groups arising from simple graphs without visual splitting, etc. This result answers positively an open question of Amrutam and Jiang from \cite{AJ22}. In the second part of the paper we obtain similar results for factors associated with groups that admit nontrivial (quasi)cohomology valued into various natural representations. In particular, we establish the ISR property for all icc, nonamenable groups that have positive first $L^2$-Betti number and contain an infinite amenable subgroup.