Word-hyperbolic group

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				|FULL LIST, MORE INFO
group satisfying a polynomial isoperimetric inequality				|FULL LIST, MORE INFO
group with solvable word problem				|FULL LIST, MORE INFO