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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.

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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.

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    $\begingroup$ You should add a reference to Olshanskiy’s paper from 1995, containing the same result. $\endgroup$ – Moishe Kohan May 25 at 13:44
  • $\begingroup$ I would be happy to do so, but what paper are you refering to ? $\endgroup$ – Mikael de la Salle May 25 at 13:51
  • $\begingroup$ SQ-universality of hyperbolic groups, Math. sbornik (1995). $\endgroup$ – Moishe Kohan May 25 at 13:56
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    $\begingroup$ The stronger result is also proved in a paper of Delzant and Gromov: doi.org/10.1112/jtopol/jtn023 $\endgroup$ – Ian Agol May 25 at 16:11
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    $\begingroup$ I believe that the correct reference to Ol'shanskii's work should be A.Yu. Ol'shanskii, On residualing homomorphisms and G-subgroups of hyperbolic groups. Internat. J. Algebra and Comput. 3 (1993), no. 4, 365--409. link $\endgroup$ – Ashot Minasyan May 26 at 20:21

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