Geodesic
Periodic factor of hyperbolic groups
Sbornik. Mathematics, Tome 72 (1992) no. 2, pp. 519-541 
A. Yu. Ol'shanskii. Periodic factor of hyperbolic groups. Sbornik. Mathematics, Tome 72 (1992) no. 2, pp. 519-541. http://geodesic.mathdoc.fr/item/SM_1992_72_2_a14/
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Voir la notice de l'article provenant de la source Math-Net.Ru

It is proved that for any noncyclic hyperbolic torsion-free group $G$ there exists an integer $n(G)$ such that the factor group $G/G^n$ is infinite for any odd $n\geqslant n(G)$. In addition, $\bigcap_{i=1}^\infty G^i=\{1\}$. (Here $G^i$ is the subgroup generated by the $i$th powers of all elements of the groups $G$.)

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