An element $g\in G$ in a group $G$ is called <b>infinitely divisible</b> if $b=y^n$ for  infinitely many different $n\in {\Bbb Z}$. It is not hard to find a finite CW-complex (or even a compact manifold) with a fundamental group containing an infinitely divisible element. For example, consider a group generated by $x$ and $b$ with a relation $xbx^{-1}=b^2$. Then $b$ is infinitely divisible. However, in a hyperbolic manifold every element can be represented by a unique shortest geodesic, which implies that infinite divisibility does not occur. Now, suppose that $G$ is a finitely generated Gromov hyperbolic group (a posteriori, it is finitely presented, as Gromov proved). It seems that it cannot contain infinitely divisible elements of infinite order. I would be very grateful for any reference to this statement.