  We study product set growth in groups with acylindrical actions on
quasi-trees and, more generally, hyperbolic spaces. As a consequence, we show
that for every surface $S$ of finite type, there exist $\alpha,\beta>0$ such
that for any finite symmetric subset $U$ of the mapping class group $MCG(S)$ we
have $|U|\geqslant (\alpha|U|)^{\beta n}$, so long as no finite index subgroup
of $\langle U\rangle$ has an infinite centre. This gives us a dichotomy for the
finitely generated subgroups of mapping class groups, which extends to virtual
subgroups.
  As right-angled Artin groups embed as subgroups of mapping class groups, this
result applies to them, and so also applies to finitely generated virtually
special groups. We separately prove that we can quickly generate loxodromic
elements in right-angled Artin groups, which by a result of Fujiwara shows that
the set of growth rates for many of their subgroups are well-ordered.
