Profinite rigidity of lamplighter groups

TL;DR

This paper proves that lamplighter groups formed by wreath products of finite cyclic groups with the integers are uniquely determined by their profinite completions, establishing their profinite rigidity.

Contribution

It demonstrates for the first time that certain lamplighter groups are profinitely rigid, expanding understanding of their algebraic and topological properties.

Findings

Lamplighter groups are profinitely rigid.

Profinite completions uniquely determine these groups.

Results apply to groups with prime cyclic factors.

Abstract

We show that the lamplighter groups $(\mathbb{Z}/p\mathbb{Z})^n\wr\mathbb{Z}$, where $p$ is prime and $n\ge 1$ is a positive integer, are profinitely rigid.