Invariant subalgebras of von Neumann algebras arising from negatively curved groups

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 Γ its von Neumann algebra L(Γ) satisfies the so-called ISR property: any von Neumann subalgebra N ⊆ L(Γ) that is normalized by all group elements in Γ is of the form N = L(Σ) for a normal subgroup Σ ⊳ Γ. In particular, this applies to all groups Γ 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 [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.

Subalgebras, subgroups, and singularity

“…□ Remark 3.2. We can also deduce the coamenability of the 𝐶 * -algebra case by arguing similarly as in the proof of [9,Corollary 5.7]. We include the proof that was kindly provided to us by the anonymous reviewer.…”

On invariant subalgebras of group and von Neumann algebras

“…In [AJ22], the authors prove a similar result as our Theorem 1.1 for a large class of 'negatively curved' groups, including all torsion free non-amenable hyperbolic groups. Their result was then generalized to all acylindrically hyperbolic groups in [CDS22].…”