Journal of the European Mathematical Society


Full-Text PDF (132 KB) | Metadata | Table of Contents | JEMS summary
Volume 21, Issue 10, 2019, pp. 3191–3197
DOI: 10.4171/JEMS/900

Published online: 2019-06-19

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

Laurent Bartholdi[1]

(1) Georg-August-Universität Göttingen, Germany

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.

Keywords: Cellular automata, Moore–Myhill theorem, amenability of groups, Ore domains, localization

Bartholdi Laurent: Amenability of groups is characterized by Myhill's Theorem. With an appendix by Dawid Kielak. J. Eur. Math. Soc. 21 (2019), 3191-3197. doi: 10.4171/JEMS/900