Surjunctive group

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 $$S$$:

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.

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.