# Word-hyperbolic group

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## Definition

A word-hyperbolic group is a finitely generated group $G$ such that the following equivalent conditions hold:

1. For any finite generating set $S$ for $G$, there exists a value $\delta > 0$ such that the word metric for $(G,S)$ is $\delta$-hyperbolic, i.e., for any triangle, each side is contained in the union of the $\delta$-neighborhoods of the other two sides.
2. There exists a finite generating set $S$ for $G$ and a value $\delta > 0$ such that the word metric for $(G,S)$ is $\delta$-hyperbolic, i.e., for any triangle, each side is contained in the union of the $\delta$-neighborhoods of the other two sides.
3. The group is a finitely presented group and has a linear isoperimetric function.
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
finitely generated free group |FULL LIST, MORE INFO
finite 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 Automatic group|FULL LIST, MORE INFO
group satisfying a polynomial isoperimetric inequality Automatic group|FULL LIST, MORE INFO
group with solvable word problem Automatic group, Group with polynomial-time solvable word problem|FULL LIST, MORE INFO