SFT covers for actions of the first Grigorchuk group

Abstract: We study symbolic dynamical representations of actions of the first Grigorchuk group $G$, namely its action on the boundary of the infinite rooted binary tree, its representation in the topological full group of a minimal substitutive $\mathbb{Z}$-shift, and its representation as a minimal system of Schreier graphs. We show that the first system admits an SFT cover, and the latter two systems are conjugate to sofic subshifts on $G$, but are not of finite type.