On the topological full group containing the Grigorchuk group

TL;DR

This paper studies the topological full group associated with a specific substitution subshift, demonstrating its finite generation and providing a simple generating set, with the Grigorchuk group embedded within it.

Contribution

It introduces a finitely generated topological full group for a particular substitution subshift and explicitly describes its generators, highlighting the embedding of the Grigorchuk group.

Findings

The topological full group is finitely generated.

A simple generating set for the group is provided.

The Grigorchuk group embeds into this full group.

Abstract

We consider the topological full group of a substitution subshift induced by a substitution $a\to aca$, $b\to d$, $c\to b$, $d\to c$. This group is interesting since the Grigorchuk group naturally embeds into it. We show that the topological full group is finitely generated and give a simple generating set for it.