Twisted Conjugacy in Direct Products of Groups

TL;DR

This paper studies the Reidemeister number and spectrum related to twisted conjugacy in direct product groups, analyzing how properties of factors influence the overall structure.

Contribution

It provides new insights into Reidemeister numbers and spectra for direct products, linking properties of individual groups to the combined structure.

Findings

Characterization of Reidemeister spectrum for direct products

Relations between factors' properties and the spectrum

Conditions under which Reidemeister numbers are finite or infinite

Abstract

Given a group $G$ and an endomorphism $\varphi$ of $G$, two elements $x, y \in G$ are said to be $\varphi$-conjugate if $x = gy \varphi(g)^{-1}$ for some $g \in G$. The number of equivalence classes for this relation is the Reidemeister number $R(\varphi)$ of $\varphi$. The set $\{R(\psi) \mid \psi \in \mathrm{Aut}(G)\}$ is called the Reidemeister spectrum of $G$. We investigate Reidemeister numbers and spectra on direct products of finitely many groups and determine what information can be derived from the individual factors.