# Surjunctive group

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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 injective 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 $h \mapsto \theta(g^{-1}h)$.
• injective if it is injective as a set map, i.e., no two elements of $S^G$ get mapped to the same element.

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

## 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.