Growth of Schreier graphs of automaton groups

Every automaton group naturally acts on the space $X^{\omega }$ of infinite sequences over some alphabet $X$. For every $w\in X^{\omega }$ we consider the Schreier graph ${\mathrm{\Gamma }}_w$ of the action of the group on the orbit of $w$. We prove that for a large class of automaton groups all Schreier graphs ${\mathrm{\Gamma }}_w$ have subexponential growth bounded above by $n^{(\log n{)}^m}$ with some constant $m$. In particular, this holds for all groups generated by automata with polynomial activity growth (in terms of S.Sidki), confirming a conjecture of V.Nekrashevych. We present applications to omega-periodic graphs and Hanoi graphs.