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 patt...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:Journal of the European Mathematical Society : JEMS 2019-06, Vol.21 (10), p.3191-3197
1. Verfasser: Bartholdi, Laurent
Format: Artikel
Sprache:eng
Schlagworte:
Online-Zugang:Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
Beschreibung
Zusammenfassung: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.
ISSN:1435-9855
1435-9863
DOI:10.4171/JEMS/900