# Understanding torsion from a presentation

Let $F_2 = \langle a,b \rangle$ be the free group on two generators, and for each word $w \in F_2$, let $G(w) = \langle a, b \ | \ w \rangle$. Is the following statement true?

$G(w)$ is torsion free if and only if for all $k \geq 2$ and for all $v \in F_2$, $w \neq v^k$

In other words, is it true that $G(w)$ is torsion free unless there is an obvious reason why it is not?

-