Universal groups of cellular automata

TL;DR

This paper explores the structure of groups generated by reversible cellular automata, revealing universal properties, classifications based on alphabet size, and connections to various algebraic groups.

Contribution

It demonstrates that the group of reversible cellular automata contains a subgroup capable of representing all finitely generated RCA groups, with detailed classifications based on alphabet size.

Findings

For prime alphabets, the group is virtually cyclic.

For composite alphabets, the group is non-amenable.

For alphabet size four, it is a linear group.

Abstract

We prove that the group of reversible cellular automata (RCA), on any alphabet $A$, contains a subgroup generated by three involutions which contains an isomorphic 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 partial 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.