Titlebook: Cellular Automata and Groups; Tullio Ceccherini-Silberstein,Michel Coornaert Textbook 2023Latest edition The Editor(s) (if applicable) and

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Surjunctive Groups,Surjunctive groups are defined in Sect. 3.1 as being the groups on which all injective cellular automata with finite alphabet are surjective. In Sect. 3.2 it is shown that every subgroup of a surjunctive group is a surjunctive group and that every locally surjunctive group is surjunctive.

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Amenable Groups,This chapter is devoted to the class of amenable groups. This is a class of groups which plays an important role in many areas of mathematics such as ergodic theory, harmonic analysis, representation theory, dynamical systems, geometric group theory, probability theory and statistics.

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Finitely Generated Groups,This chapter is devoted to the growth and amenability of finitely generated groups. The choice of a finite symmetric generating subset for a finitely generated group defines a word metric on the group and a labelled graph, which is called a Cayley graph.

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Cellular Automata,n is defined as being a map from the group into the alphabet. Thus, a configuration is a way of attaching an element of the alphabet to each element of the group. There is a natural action of the group on the set of configurations which is called the shift action (see Sect. 1.1).

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Linear Cellular Automata,induced vector space structure on the set of configurations. If the alphabet vector space and the underlying group are fixed, the set of linear cellular automata is a subalgebra of the endomorphism algebra of the configuration space (Proposition 8.1.4).

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