Sofic 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
The definition provides insufficient context, such as defining terms that are not clearly explained (either directly or through links to their own pages).
A sofic group is a group satisfying the following equivalent conditions:
- Its Cayley graph is initially subamenable.
- It is a subgroup of an ultraproduct of symmetric groups on finite sets with the property that any two elements of the group have distance 1.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| finite group | finite implies sofic | |FULL LIST, MORE INFO | ||
| residually finite group | Template:Intermediate notion short | |||
| amenable discrete group | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| surjunctive group | (see page), note that surjunctivity conjecture states that every group is surjunctive. | sofic implies surjunctive | |FULL LIST, MORE INFO |