  The main theorem in this paper is that for any group $G$ that acts
acylindrically on a quasi-tree, there exists $\alpha>0$ such that for any
finite $U \subset G$ that has large enough displacement in the quasi-tree, and
is not contained in a virtually cyclic subgroup, we have $|U^n|>(\alpha
|U|)^{\lfloor \frac{n+1}{2}\rfloor}$. By a result of Balasubramanya, the main
theorem applies to every acylindrically hyperbolic group. In particular, we can
show that if $G$ is a right-angled Artin group, then there exist
$\alpha,\beta>0$ such that for any finite symmetric $U\subset G$ that is not
contained in $\mathbb{Z}$, and not contained non-trivially in a subgroup of the
form $H\times\mathbb{Z}$, we have $|U^n|\geqslant(\alpha |U|)^{\beta n}$.
