hyperbolic quotient of hyperbolic group

I have a memory of hearing about a result (or perhaps a conjecture), possibly due to Gromov, that, if $$G$$ is a hyperbolic group and $$g \in G$$ has infinite order, then the quotient group $$G/\langle (g^n)^G \rangle$$ is hyperbolic for all sufficiently large $$n > 0$$.

I have been searching for references, but without success. Can anyone help?.

$$\mathbf{Edit}$$: After looking at the references in the answer by Mikael de la Salle, I see that I did not state this result correctly. Rather than the statement being for all sufficiently large $$n>0$$, it should be that there exists and $$N>0$$ such that $$G/\langle (g^{nN})^G \rangle$$ is hyperbolic for all $$n > 0$$. The result stated applies only to non-elementary hyperbolic groups, but for an elementary hyperbolic group this quotient is finite, and so it remains correct.