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 ${\exp }^k\left(n^{p/q}\right)$. Here, ${\exp }^k$ denotes the k-fold-iterated exponential function.