Veelike actions and the MCG of a mixing SFT

We embed Thompson's group $V$ in the mapping class group of a mixing subshift of finite type. Question~6.3 in [Boyle-Chuysurichay, 17] asks whether these mapping class groups are sofic. Our result suggests that this question is difficult to solve at present (at least for some mixing SFTs), since a resolution of it in either direction would solve an open problem in geometric group theory. More generally, we define the notion of a ``veelike action'', and prove that whenever a group acts veelike on a language, it embeds in the mapping class group of some subshift. We show that Thompson's $V$ acts veelike on a locally testable language, and thus it embeds in the mapping class group of a mixing SFT. A two-sided variant of the argument works for the Brin-Thompson group $2V$.