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

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.