20.39 (2022)

Let $F$ be a non-abelian finitely generated free group, $1 \neq w \in F$, and $n \geqslant 1$. Is the group $\langle F, t \mid t^n = w \rangle$ linear of degree 2 over a field of characteristic 0 if $w$ is not a proper power in $F$?

This question is motivated by the following well-known question: is it true that the free $\mathbb{Q}$-group $F^{\mathbb{Q}}$ is linear over a field? (See 13.39(b).)