A1 Refereed original research article in a scientific journal
Universal groups of cellular automata
Authors: Salo Ville
Publisher: ARS POLONA-RUCH
Publication year: 2022
Journal: Colloquium Mathematicum
Journal name in source: COLLOQUIUM MATHEMATICUM
Journal acronym: COLLOQ MATH-WARSAW
Number of pages: 39
ISSN: 0010-1354
eISSN: 1730-6302
DOI: https://doi.org/10.4064/cm8368-9-2021
Web address : https://www.impan.pl/en/publishing-house/journals-and-series/colloquium-mathematicum/all///114528/universal-groups-of-cellular-automata
Self-archived copy’s web address: https://research.utu.fi/converis/portal/detail/Publication/174837394
We prove that the group of reversible cellular automata (RCA), on any alphabet A, contains a subgroup generated by three involutions which contains an iso-morphic copy of every finitely generated group of RCA on any alphabet B. This result follows from a case study of groups of RCA generated by symbol permutations and par-tial shifts (equivalently, partitioned cellular automata) with respect to a fixed Cartesian product decomposition of the alphabet. For prime alphabets, we show that this group is virtually cyclic, and that for composite alphabets it is non-amenable. For alphabet size four, it is a linear group. For non-prime non-four alphabets, it contains copies of all finitely generated groups of RCA. We also prove this property for the group generated by RCA of biradius one on any full shift with large enough alphabet, and also for some perfect finitely generated groups of RCA.
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