Twisted conjugacy in direct products of groups

Abstract

Given a group G and an endomorphism $\phi$ of G, two elements $x,y\in G$ are said to be $\phi$-conjugate if $x=gy\phi {(g)}^{-1}$ for some $g\in G.$ The number of equivalence classes for this relation is the Reidemeister number $R(\phi )$ of $\phi .$ The set $\{R(\psi )|\psi \in \text{Aut}(G)\}$ 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.

Keywords:

• Direct product
• ${\mathit{R}}_{\infty }$-property
• Reidemeister number
• twisted conjugacy

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 20E36
• 20E45
• Secondary: 20E22

Acknowledgments

The author thanks Karel Dekimpe and the anonymous referee for their useful remarks and suggestions, in particular for their examples of concrete groups to which the results apply.

Additional information

Funding

This work was supported by the Research Foundation – Flanders (FWO) under Grant 1112520N.