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. 1 Actions on rooted trees, automaton groups, and Hanoi Towers groups The free monoid X of words over the alphabet X = {0, . . . , k−1} ordered by the prefix relation has a kregular rooted tree structure in which the empty word is the root and the words of length n constitute the level n in the tree. The k children of the vertex (word) u are the vertices (words) ux, for x = 0, . . . , k−1. Denote this k-regular rooted tree by T . Any automorphism g of the tree T can be (uniquely) decomposed as g = πg (g0, g1, . . . , gk−1), where πg is a permutation in SX = Sk, called the root permutation of g, and gx, x = 0, . . . , k − 1, are tree automorphisms, called the (first level) sections of g. The root permutation πg and the sections gi are determined uniquely by the relation g(xw) = πg(x)gx(w), for all x ∈ X and w ∈ X . The action of a tree automorphism can be extended to an isometric action on the boundary ∂T consisting of the infinite words over X . The space ∂T is a compact ultrametric space homeomorphic to a Cantor set. For any permutation π in Sk define a k-ary tree automorphism a = aπ by a = π (a0, a1, . . . , ak−1), where ai is the identity automorphism if i is in the support of π and ai = a otherwise. The action of the automorphism a(ij) on T is given (recursively) by a(ij)(xw) = 