Free metabelian groups are permutation stable

Abstract

We prove that all finitely generated free metabelian groups are permutation stable by verifying that all invariant random subgroups of them are co-sofic. This partially answers to the question asked by Levit and Lubotzky whether all finitely generated metabelian groups are permutation stable. Our proof extends the range of application of Levit and Lubotzky’s method, which is used to show permutation stability of permutational wreath products of finitely generated abelian groups, to non-split and non-permutational metabelian groups.