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

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.