Sofic 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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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

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:

  1. Its Cayley graph is initially subamenable.
  2. 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