Conjugacy depth function for generalised lamplighter groups

TL;DR

This paper characterizes the asymptotic behavior of conjugacy separability in lamplighter groups and provides bounds for wreath products of finitely generated abelian groups with specific conditions.

Contribution

It offers a complete characterization of conjugacy separability in lamplighter groups and establishes exponential bounds for a broad class of wreath products.

Findings

Asymptotic behavior of conjugacy separability is fully characterized.

Exponential upper and lower bounds are established for wreath products.

Results apply to all wreath products of finitely generated abelian groups with infinite acting group and finite base group.

Abstract

In this article, we completely characterize the asymptotic behavior of conjugacy separability for the lamplighter groups. More generally, we give exponential upper and lower bounds for all wreath products of finitely generated abelian groups where the acting group is infinite and base group is finite.