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Profinite rigidity of lamplighter groups

Guy Blachar

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.

Profinite rigidity of lamplighter groups

Abstract

We show that the lamplighter groups , where is prime and is a positive integer, are profinitely rigid.

Paper Structure

This paper contains 7 sections, 16 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. The groups L_n,p
  3. Proof of the main theorem
  4. A semidirect product decomposition
  5. The structure of N
  6. Determining the module structure of N
  7. Conclusion of the proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

The group $\mathcal{L}_{n,p}=(\mathbb{Z}/p\mathbb{Z})^n\wr\mathbb{Z}$ is profinitely rigid for any prime $p$ and any positive integer $n\ge 1$.

Theorems & Definitions (27)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • Corollary 2.3
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • proof
  • Corollary 3.3
  • ...and 17 more