Growth of Schreier graphs of automaton groups

Every automaton group naturally acts on the space Xω of infinite sequences over some alphabet X. For every $${w \in X^{\omega}}$$ we consider the Schreier graph Γw of the action of the group on the orbit of w. We prove that for a large class of automaton groups all Schreier graphs Γw 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 ω-periodic graphs and Hanoi graphs.