Hyperbolic groups and their quotients of bounded exponents

In 1987, Gromov conjectured that for every non-elementary hyperbolic group $G$ there is an $n =n(G)$ such that the quotient group $G/G^{n}$ is infinite. The article confirms this conjecture. In addition, a description of finite subgroups of $G/G^{n}$ is given, it is proven that the word and conjugacy problem are solvable in $G/G^{n}$ and that $\bigcap _{k=1}^{\infty }G^{k} = \{ 1\}$. The proofs heavily depend upon prior authors' results on the Gromov conjecture for torsion free hyperbolic groups and on the Burnside problem for periodic groups of even exponents.