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Bounding conjugacy depth functions for wreath products of finitely generated abelian groups

Michal Ferov, Mark Pengitore

Abstract

In this article, we study the asymptotic behaviour of conjugacy separability for wreath products of abelian groups. We fully characterise the asymptotic class in the case of lamplighter groups and give exponential upper and lower bounds for generalised lamplighter groups. In the case where the base group is infinite, we give superexponential lower and upper bounds. We apply our results to obtain lower bounds for conjugacy depth functions of various wreath products of groups where the acting group is not abelian.

Bounding conjugacy depth functions for wreath products of finitely generated abelian groups

Abstract

In this article, we study the asymptotic behaviour of conjugacy separability for wreath products of abelian groups. We fully characterise the asymptotic class in the case of lamplighter groups and give exponential upper and lower bounds for generalised lamplighter groups. In the case where the base group is infinite, we give superexponential lower and upper bounds. We apply our results to obtain lower bounds for conjugacy depth functions of various wreath products of groups where the acting group is not abelian.
Paper Structure (18 sections, 36 theorems, 161 equations)

This paper contains 18 sections, 36 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Statement of the results
  4. Outline of the paper
  5. Preliminaries
  6. Asymptotic notions and depth functions
  7. Wreath products
  8. Wreath products and Laurent polynomial rings
  9. Lower bounds
  10. Lower bounds for $\mathop{\mathrm{Conj}}\nolimits_{A \wr B}(n)$ where $A$ is a finite abelian group
  11. Lower bounds for $\mathop{\mathrm{Conj}}\nolimits_{A \wr B}(n)$ when $A$ and $B$ are infinite
  12. Upper bounds
  13. Simultaneous cosets
  14. Translations
  15. Finite base groups
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $A$ be a finite abelian group, and suppose that $B$ is an infinite finitely generated abelian group. If the torsion free rank of $B$ is $1$, then If the torsion free rank of $B$ is $k \geq 2$, then

Theorems & Definitions (64)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 54 more