The Profinite Rigidity of Torsion-Free Lamplighter Groups

We prove that the torsion-free lamplighter group $\mathrm{\Gamma }={\mathbb{Z}}^n\wr \mathbb{Z}$ of any rank $n\in \mathbb{N}$ is profinitely rigid in the absolute sense: the finite quotients of $\mathrm{\Gamma }$ determine its isomorphism type uniquely among all finitely generated residually finite groups. The proof combines the theory of profinite rigidity for modules over Noetherian domains with an analysis of the algebraic properties of the lower central series of groups with the same profinite completion as $\mathrm{\Gamma }$.