  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 $2V$.
