Amenability of groups is characterized by Myhill's Theorem
  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. 
Laurent Bartholdi, Dawid Kielak
cs.FL math.GR