Product set growth in virtual subgroups of mapping class groups

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.