Twisted conjugacy in some lamplighter-type groups

TL;DR

This paper investigates the twisted conjugacy property in lamplighter-type groups, specifically restricted wreath products of finite abelian groups with b^k, and determines when these groups possess the $R_ infty$ property.

Contribution

It provides a near-complete classification of the $R_ infty$ property for restricted wreath products of finite abelian groups with b^k.

Findings

Most such wreath products have the $R_ infty$ property.

The paper identifies specific conditions under which the property holds or fails.

Results extend understanding of automorphism behaviors in lamplighter-type groups.

Abstract

For a restricted wreath product $G\wr \mathbb{Z}^k$, where $G$ is a finite abelian group, we determine (almost in all cases) whether this product has the $R_\infty$ property (i.e., each its automorphism has infinite Reidemeister number).