Let $G$ be a torsion-free group. For an element $g \in G \setminus \{1_G\}$ we define the radical of $g$ in $G$ as $$ \operatorname{Rad}_G(g) = \left\{r \in G \mid r^a \in \langle g \rangle \mbox{ for some } a \in \mathbb{Z}\setminus\{0\}\right\}. $$
For which torsion-free groups can we show that $\operatorname{Rad}_G(g)$ is an infinite cyclic subgroup of $G$ for every nontrivial element?
So far I have been able to establish it for the following classes:
- residually finitely generated torsion-free nilpotent groups,
- torsion-free hyperbolic groups,
- relatively hyperbolic groups (in the sense of Bowdich), where the associated subgroups already have the property (this includes for example toral relatively-hyperbolic groups).
Are there any easy examples I am missing?