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

TL;DR

This paper investigates the asymptotic behavior of conjugacy separability in wreath products of abelian groups, providing bounds and characterizations for lamplighter and generalized lamplighter groups, with implications for non-abelian cases.

Contribution

It offers a complete characterization of conjugacy separability for lamplighter groups and establishes exponential and superexponential bounds for generalized and infinite base groups.

Findings

Conjugacy separability behaves exponentially in lamplighter groups.

Superexponential bounds are established for infinite base groups.

Results provide lower bounds for conjugacy depth in non-abelian wreath products.

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.