Automatic group
Definition
An automatic group is a finitely generated group satisfying the following equivalent conditions:
- There exists a finite generating set with respect to which the group possesses an automatic structure.
- For every finite generating set, the group possesses an automatic structure with respect to that generating set.
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| biautomatic group | |FULL LIST, MORE INFO | |||
| finite group | |FULL LIST, MORE INFO | |||
| word-hyperbolic group | word-hyperbolic implies automatic | automatic not implies word-hyperbolic | |FULL LIST, MORE INFO | |
| finitely generated free group | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| group satisfying a quadratic isoperimetric inequality | |FULL LIST, MORE INFO | |||
| group satisfying a polynomial isoperimetric inequality | |FULL LIST, MORE INFO | |||
| group with polynomial-time solvable word problem | |FULL LIST, MORE INFO | |||
| group with solvable word problem | |FULL LIST, MORE INFO |