Topological full groups of minimal subshifts with subgroups of intermediate growth

Abstract

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.