Amenability of groups is characterized by Myhill's Theorem. With an appendix by Dawid Kielak

Abstract

We prove a converse to Myhill's "Garden-of-Eden" theorem and obtain in this manner a characterization of amenability in terms of cellular automata: A group $G$ is amenable if and only if every cellular automaton with carrier $G$ that has gardens of Eden also has mutually erasable patterns.

This answers a question by Schupp, and solves a conjecture by Ceccherini-Silberstein, Machì and Scarabotti.

Furthermore, for non-amenable $G$ the cellular automaton with carrier $G$ that has gardens of Eden but no mutually erasable patterns may also be assumed to be linear.

An appendix by Dawid Kielak shows that group rings without zero divisors are Ore domains precisely when the group is amenable, answering a conjecture attributed to Guba.