On invariant von Neumann subalgebras rigidity property

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 $\Lambda ⊲\Gamma$. 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.

v3: final version, accepted to JFA. v2: title changed due to the change of name "non-commutative Margulis property" to "invariant von Neumann subalgebras rigidity property"; minor revision and typos corrected