Free metabelian groups are permutation stable

TL;DR

This paper proves that all finitely generated free metabelian groups are permutation stable, extending existing methods to a broader class of metabelian groups and partially answering a longstanding question.

Contribution

It demonstrates permutation stability for all finitely generated free metabelian groups, expanding the applicability of Levit-Lubotzky's method to new classes of metabelian groups.

Findings

Finitely generated free metabelian groups are permutation stable.

Extension of Levit-Lubotzky's method to non-split, non-permutational metabelian groups.

Partial answer to the question on permutation stability of all finitely generated metabelian groups.

Abstract

We prove that all finitely generated free metabelian groups are permutation stable. 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-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.