Asymptotic aspects of Schreier graphs and Hanoi Towers groups

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Amenability of linear-activity automaton groups (2009, background): “Its level Schreier graphs are discrete Sierpinski gaskets (see Grigorchuk and Šunić, 2006).”
Milnor's Problem on the Growth of Groups and its Consequences (2011, background): “There are examples of actions with quite exotic intermediate behavior like n m n for some m > 0 [33, 114, 37, 38].”
Schreier graphs of the Basilica group (2009, unknown): “[1],[7],[12].”
Hausdorff dimension in a family of self-similar groups (2006, background): “It is known that the closureG of the Hanoi groupH on 3 pegs (see [17] for a definition of H as an automaton group) is finitely constrained group [16] with forbidden patterns of size 2.”