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.



hyperbolic quotient of hyperbolic group
This is contained in at least Delzant's paper Sous-groupes distingués et quotients des groupes hyperboliques. [Distinguished subgroups and quotients of hyperbolic groups] Duke Mathematical Journal, vol. 83 (1996), no. 3, pp. 661–682, and also in Ol'shanskii's paper SQ-universality of hyperbolic groups, Mat. Sb. 186 (1995), no. 8, 119–132.


I am not an expert, but the first lines of the papers seem to indicate that this result was announced by Gromov, but that the proofs were not all convicing.


