Surjunctive group
From Groupprops
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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Contents
Definition
Definition in terms of equivariant functions between function spaces
Suppose is a group. We say that is a surjunctive group if the following holds for every finite set :
Consider the set of all functions from to , and wndow this set with a topology (using the product topology from the discrete topology on ). Then, any continuous -equivariant injective function from must be surjective.
Terminology: Call a function :
- continuous if it is continuous with respect to the topology specified above (the product topology arising from the discrete topology on .
- -equivariant if for all , where is defined as the function .
- injective if it is injective as a set map, i.e., no two elements of get mapped to the same element.
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 is a group such that every finitely generated subgroup of is surjunctive, then 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.