Amenability of quadratic automaton groups

TL;DR

This paper establishes the amenability of quadratic automaton groups by deriving new resistance bounds on Schreier graphs, using a novel weighted Nash-Williams criterion.

Contribution

It introduces a new weighted Nash-Williams criterion and applies it to prove amenability of quadratic automaton groups.

Findings

Lower bounds for electrical resistance in Schreier graphs of quadratic automaton groups.

Proof that all quadratic activity automaton groups are amenable.

Development of a new weighted Nash-Williams criterion for resistance bounds.

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 zero ray. These bounds, combined with results of \cite{JNS} 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.