On invariant von Neumann subalgebras rigidity property

We say that a countable discrete group Γ satisfies the invariant von Neumann subalgebras rigidity (ISR) property if every Γ- invariant von Neumann subalgebra M in L(Γ) is of the form L(Λ) for some normal subgroup Λ◁Γ. We show many “negatively curved” groups, including all torsion free non-amenable hyperbolic groups and torsion free groups with positive first L²-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.