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Grigorchuk's group and aperiodic order

We discuss Grigorchuk’s group of intermediate growth, its natural action on a binary rooted tree and its substitution structure. We then present a construction of Grigorchuk, Lenz and Nagnibeda which relates the associated measurable dynamical system of graphs and its associated random Laplacian to a combinatorial dynamical system over the integers and an associated random Schrödinger operator.