Asymptotic aspects of Schreier graphs and Hanoi Towers groups
  We present relations between growth, growth of diameters and the rate of vanishing of the spectral gap in Schreier graphs of automaton groups. In particular, we introduce a series of examples, called Hanoi Towers groups since they model the well known Hanoi Towers Problem, that illustrate some of the possible types of behavior. 
Rostislav Grigorchuk and Zoran Sunik