  We show that every Grigorchuk group $G_\omega$ embeds in (the commutator
subgroup of) the topological full group of a minimal subshift. In particular,
the topological full group of a Cantor minimal system can have subgroups of
intermediate growth, a question raised by Grigorchuk; it can also have finitely
generated infinite torsion subgroups, as well as residually finite subgroups
that are not elementary amenable, answering questions of Cornulier. By
estimating the word-complexity of this subshift, we deduce that every
Grigorchuk group $G_\omega$ can be embedded in a finitely generated simple
group that has trivial Poisson boundary for every simple random walk.
