Surjunctive group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition

Definition in terms of equivariant functions between function spaces

Suppose $G$ is a group. We say that $G$ is a surjunctive group if the following holds for every finite set $S$:

Consider the set $S^G$ of all functions from $G$ to $S$, and wndow this set with a topology (using the product topology from the discrete topology on $S$). Then, any continuous $G$-equivariant function from $S^G \to S^G$ must be surjective.

Terminology: Call a function $f:S^G \to S^G$:

• continuous if it is continuous with respect to the topology specified above (the product topology arising from the discrete topology on $S$.
• $G$-equivariant if $f(g \cdot \theta) = g \cdot f(\theta)$ for all $\theta in S^G$, where $g \cdot \theta$ is defined as the function $h \mapsto \theta(g^{-1}h)$.

The surjunctivity conjecture (currently open) states that every group is surjunctive.

Definition in terms of cellular automata

See the Wikipedia page.

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
local group property	Yes	surjunctivity is local	If $G$ is a group such that every finitely generated subgroup of $G$ is surjunctive, then $G$ itself is surjunctive.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
finite group		finite implies surjunctive	(follows from any of the intermediate properties listed)	Sofic group|FULL LIST, MORE INFO
residually finite group	intersection of subgroups of finite index is trivial	residually finite implies surjunctive		Sofic group|FULL LIST, MORE INFO
locally residually finite group	every finitely generated subgroup is residually finite.	locally residually finite implies surjunctive	any abelian group that is not residually finite	|FULL LIST, MORE INFO
abelian group	any two elements commute	abelian implies surjunctive	any finite non-abelian group	|FULL LIST, MORE INFO
sofic group	its Cayley graph is initially subamenable	sofic implies surjunctive	any abelian non-sofic group	Template:Interemdiate notions short

Weaker properties

The (currently open) surjunctivity conjecture states that every group is surjunctive. This is equivalent to asserting that surjunctivity is the tautology, i.e., the weakest possible group property. Since the conjecture is open, there is no known group property other than the tautology that is weaker than surjunctivity.