Permutation stability of Grigorchuk groups

A recent result of Becker, Lubotzky and Thom characterizes, for amenable groups, permutation stability in terms of co-soficity of invariant random subgroups (IRS). We will explain that for a class of amenable groups acting on rooted trees, including the Grigorchuk group, the IRS co-soficity condition is verified. One key ingredient in the proof is the so called “double commutator” lemma for IRS, which is an analogue of the classical lemma known for normal subgroups. All notions will be defined and explained.