Growth of Schreier graphs of automaton groups

TL;DR

This paper proves that Schreier graphs of a broad class of automaton groups exhibit subexponential growth, specifically bounded by a polynomial of a logarithmic power, confirming a conjecture and applying to various graph types.

Contribution

It establishes subexponential growth bounds for Schreier graphs of automaton groups with polynomial activity, confirming Nekrashevych's conjecture.

Findings

Schreier graphs have subexponential growth bounded by n^{(log n)^m}

All groups generated by automata with polynomial activity growth satisfy this bound

Applications include omega-periodic and Hanoi graphs

Abstract

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 $\Gamma_w$ of the action of the group on the orbit of $w$. We prove that for a large class of automaton groups all Schreier graphs $\Gamma_w$ 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.