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.