Amenability of groups is characterized by Myhill's theorem

Bartholdi L, Kielak D (2019)
Journal of the European Mathematical Society 21(10): 3191-3197.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Autor/in
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Abstract / Bemerkung
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, Machi 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.
Stichworte
Cellular automata; Moore-Myhill theorem; amenability of groups; Ore; domains; localization
Erscheinungsjahr
2019
Zeitschriftentitel
Journal of the European Mathematical Society
Band
21
Ausgabe
10
Seite(n)
3191-3197
ISSN
1435-9855
Page URI
https://pub.uni-bielefeld.de/record/2937122

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Bartholdi L, Kielak D. Amenability of groups is characterized by Myhill's theorem. Journal of the European Mathematical Society. 2019;21(10):3191-3197.
Bartholdi, L., & Kielak, D. (2019). Amenability of groups is characterized by Myhill's theorem. Journal of the European Mathematical Society, 21(10), 3191-3197. doi:10.4171/JEMS/900
Bartholdi, L., and Kielak, D. (2019). Amenability of groups is characterized by Myhill's theorem. Journal of the European Mathematical Society 21, 3191-3197.
Bartholdi, L., & Kielak, D., 2019. Amenability of groups is characterized by Myhill's theorem. Journal of the European Mathematical Society, 21(10), p 3191-3197.
L. Bartholdi and D. Kielak, “Amenability of groups is characterized by Myhill's theorem”, Journal of the European Mathematical Society, vol. 21, 2019, pp. 3191-3197.
Bartholdi, L., Kielak, D.: Amenability of groups is characterized by Myhill's theorem. Journal of the European Mathematical Society. 21, 3191-3197 (2019).
Bartholdi, Laurent, and Kielak, Dawid. “Amenability of groups is characterized by Myhill's theorem”. Journal of the European Mathematical Society 21.10 (2019): 3191-3197.