SFT covers for actions of the first Grigorchuk group

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 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.