Fractional distortion in hyperbolic groups

For all integers $p>q>0$ and $k>0$, and all non-elementary torsion-free hyperbolic groups $H$, we construct a hyperbolic group $G$ in which $H$ is a subgroup, such that the distortion function of $H$ in $G$ grows like ${\exp }^k(n^{p\mathrm{/}q})$. Here, ${\exp }^k$ denotes the $k$-fold-iterated exponential function.