Amenability of quadratic automaton groups

Abstract

We give lower bounds for the electrical resistance between vertices in the Schreier graphs of the action of the linear (degree 1) and quadratic (degree 2) mother groups on the orbit of the 0-ray. These bounds, combined with results of Juschenko et al. (2016), show that every quadratic activity automaton group is amenable. The resistance bounds use an apparently new “weighted” version of the Nash-Williams criterion which may be of independent interest.