| Summary: | <p>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.</p>
<p>This answers a question by Schupp, and solves a conjecture by Ceccherini-Silberstein, Machì and Scarabotti.</p>
<p>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.</p>
<p>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.</p>
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