Amenability of groups is characterized by Myhill's Theorem

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. 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.

2nd version including results on Ore domains

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• Garden of Eden and Specification
• Improved algebraic fibrings
• A Garden of Eden theorem for principal algebraic actions
• The Garden of Eden theorem: old and new
• Pre-Expansivity in Cellular Automata
• Dynamical correspondences of $L^2$-Betti numbers
• Entropy on modules over the group ring of a sofic group