Topological full groups of minimal subshifts with subgroups of intermediate growth

Abstract

This work is partially supported by the ERC starting grant GA 257110 “RaWG”.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; moreover it can have finitely generated infinite torsion subgroups, answering a question 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. This work is partially supported by the ERC starting grant GA 257110 “RaWG”.