Amenability of groups is characterized by Myhill's Theorem

TL;DR

This paper characterizes amenability of groups through cellular automata, establishing a new equivalence involving gardens of Eden and mutually erasable patterns, and resolves several longstanding conjectures.

Contribution

It provides a novel characterization of group amenability via cellular automata properties, answering key open questions in the field.

Findings

A group is amenable iff certain cellular automata with gardens of Eden have mutually erasable patterns.

Proves a converse to Myhill's Garden-of-Eden theorem.

Shows group rings without zero divisors are Ore domains iff the group is amenable.

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, Mach\`i and Scarabotti. An appendix by Dawid Kielak proves that group rings without zero divisors are Ore domains precisely when the group is amenable, answering a conjecture attributed to Guba.