On invariant von Neumann subalgebras rigidity property

TL;DR

This paper introduces the invariant von Neumann subalgebras rigidity (ISR) property, showing that many negatively curved groups have this property, which constrains their invariant subalgebras to be associated with normal subgroups.

Contribution

The paper establishes the ISR property for a broad class of negatively curved groups, including hyperbolic groups and certain products, advancing understanding of their algebraic rigidity.

Findings

Many negatively curved groups satisfy the ISR property.

Hyperbolic groups and certain product groups have invariant von Neumann subalgebras of a specific form.

The paper discusses potential relaxation of the torsion-free assumption.

Abstract

We say that a countable discrete group $\Gamma$ satisfies the invariant von Neumann subalgebras rigidity (ISR) property if every $\Gamma$- invariant von Neumann subalgebra $\mathcal{M}$ in $L(\Gamma)$ is of the form $L(\Lambda)$ for some normal subgroup $\Lambda\lhd \Gamma$. We show many ``negatively curved" groups, including all torsion free non-amenable hyperbolic groups and torsion free groups with positive first $L^2$-Betti number under a mild assumption, and certain finite direct product of them have this property. We also discuss whether the torsion-free assumption can be relaxed.