Reidemeister spectrum for metabelian groups of the form ${Q}^n\rtimes \mathbb Z$ and ${\mathbb Z[1/p]}^n\rtimes \mathbb Z$, $p$ prime

TL;DR

This paper investigates the Reidemeister spectrum of specific metabelian groups formed by semidirect products involving rational and p-adic integers, focusing on the $R_{ty}$ property and introducing a Nielsen spectrum concept.

Contribution

It provides a detailed analysis of the Reidemeister spectrum for groups of the form ${f Q}^n times f Z$ and ${f Z}[1/p]^n times f Z$, and explores the $R_{ty}$ property within these groups.

Findings

Characterization of the Reidemeister spectrum for the groups studied.

Identification of conditions for the $R_{ty}$ property.

Introduction of the Nielsen spectrum concept and examples.

Abstract

In this note we study the Reidemeister spectrum for metabelian groups of the form ${\mathbb Q}^n\rtimes \mathbb Z$ and ${\mathbb Z[1/p]}^n\rtimes \mathbb Z$. Particular attention is given to the $R_{\infty}$ property of a subfamily of these groups. We also define a Nielsen spectrum of a space and discuss some examples.