Product set growth in groups and hyperbolic geometry

Generalising results of Razborov and Safin, and answering a question of Button, we prove that for every hyperbolic group there exists a constant $\alpha >0$ such that for every finite subset $U$ that is not contained in a virtually cyclic subgroup $|U^n|\geqslant (\alpha |U|)^{[(n+1)/2]}$. Similar estimates are established for groups acting acylindrically on trees or hyperbolic spaces.