The Profinite Rigidity of Free Metabelian Groups

Abstract: We prove that finitely generated free metabelian groups $\Psi_n$ are profinitely rigid in the absolute sense: they are distinguished by their finite quotients among all finitely generated residually finite groups. The proof is based on a previous result of the author governing profinite rigidity for modules over Noetherian domains, as well as a homological characterisation of free metabelian groups due to Groves--Miller.