The Profinite Rigidity of Torsion-Free Lamplighter Groups

TL;DR

This paper establishes that torsion-free lamplighter groups are uniquely determined by their finite quotients, demonstrating their profinite rigidity through algebraic and module-theoretic analysis.

Contribution

It proves the profinite rigidity of torsion-free lamplighter groups of any rank, a significant advancement in understanding their algebraic structure.

Findings

Finite quotients determine the isomorphism type of the group

Profinite rigidity holds for all ranks of lamplighter groups

Analysis combines module theory and lower central series properties

Abstract

We prove that the torsion-free lamplighter group $\Gamma = \mathbb{Z}^n \wr \mathbb{Z}$ of any rank $n \in \mathbb{N}$ is profinitely rigid in the absolute sense: the finite quotients of $\Gamma$ 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 $\Gamma$.