Let $\Gamma$ be a nonelementary hyperbolic group. Then is it true that $\Gamma^{p}:=<\gamma^{p}| \gamma \in \Gamma>$ is a finite index subgroup of $\Gamma$? Here $p$ is a prime number. What is known about this problem? What can we say when $p$ is just an odd number(not necessarily prime) or just a postive integer?



Burnside problem for hyperbolic groups?
A. Yu. Olshanskii in the paper "Periodic quotient groups of hyperbolic groups." ((Russian) Mat. Sb. 182 (1991), no. 4, 543--567; translation in Math. USSR-Sb. 72 (1992), no. 2, 519–541) proved that for every torsion-free non-elementary hyperbolic group $G$ there is a number $N \in \mathbb{N}$ such that for any odd $n \ge N$ the quotient $G/G^n$ is infinite. 


In a more recent article, Ivanov and Olshanskii ("Hyperbolic groups and their quotients of bounded exponents". Trans. Amer. Math. Soc. 348 (1996), no. 6, 2091–2138) proved a similar statement for an arbitrary non-elementary hyperbolic group $G$ (torsion is allowed): there is $n=n(G) \in \mathbb{N}$ such that $G/G^n$ is infinite. In this case one cannot say that $G/G^k$ is infinite for any sufficiently large odd $k$, because if the group $G$ is generated by elements of, say, order $3$, then for any $k$ not divisible by $3$, $G^k=G$.


