Topological full groups of minimal subshifts with subgroups of intermediate growth

TL;DR

This paper demonstrates that topological full groups of minimal subshifts can embed complex subgroups like Grigorchuk groups, exhibiting intermediate growth, torsion, and non-elementary amenability, thus answering several open questions.

Contribution

It shows that topological full groups of minimal subshifts can contain subgroups with intermediate growth and other complex properties, expanding understanding of their structure.

Findings

Embedding of Grigorchuk groups into topological full groups

Existence of finitely generated simple groups with trivial Poisson boundary

Presence of subgroups with intermediate growth and torsion properties

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.