Surjunctive Groups

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Abstract

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. Every locally residually finite group is surjunctive (Corollary 3.3.6). The class of locally residually finite groups is quite large and includes in particular all finite groups, all abelian groups, and all free groups (a still wider class of surjunctive groups, namely the class of sofic groups, will be described in Chap. 7). In Sect. 3.4, given an arbitrary group Γ, we introduce a natural topology on the set of its quotient groups. In Sect. 3.7, it is shown that the set of surjunctive quotients is closed in the space of all quotients of Γ.