The $R_\infty$ property for free groups, free nilpotent groups and free solvable groups

TL;DR

This paper characterizes which free nilpotent and free solvable groups possess the $R_ ext{infty}$ property, and establishes that free groups of infinite rank do not have this property.

Contribution

It provides a complete classification of the $R_ ext{infty}$ property for free nilpotent and free solvable groups, and clarifies the case of infinite rank free groups.

Findings

Free nilpotent and free solvable groups have the $R_ ext{infty}$ property under specific conditions.

Free groups of infinite rank do not have the $R_ ext{infty}$ property.

The paper precisely determines which groups in these classes possess the $R_ ext{infty}$ property.

Abstract

Let $F$ be either a free nilpotent group of a given class and of finite rank or a free solvable group of a certain derived length and of finite rank. We show precisely which ones have the $R_{\infty}$ property. Finally, we also show that the free group of infinite rank does not have the $R_{\infty}$ property.