In group theory, a hyperbolic group, also called negatively curved group, word-hyperbolic group, Gromov-hyperbolic group, δ-hyperbolic group, is a finitely generated group equipped with a word metric satisfying certain properties characteristic of hyperbolic geometry.
There are several equivalent definitions. The first is the so-called thin triangles condition, generally credited to Eliyahu Rips. Let G be a finitely generated group, and T be its Cayley graph with respect to a set of generators. By identifying each edge isometrically with the unit interval in 
, we can define a metric on T by defining the distance between each pair of points x and y in T to be the minimum length over all paths from x to y. A shortest path between two points is called a geodesic segment.
A triangle in T is simply three points (the vertices) with each pair being joined by a geodesic segment (a side). Let 
. A triangle is δ-thin if each side is contained in a δ-neighborhood of the other two sides. If every triangle in T is δ-thin, then we say G is δ-hyperbolic. This condition is actually a quasi-isometric invariant, so in particular, does not depend on the set of generators chosen (although the actual value for δ may change).
By imposing this condition on geodesic metric spaces in general, we arrive at the more general notion of δ-hyperbolic space.
References
Mikhail Gromov, Hyperbolic groups. Essays in group theory, 75--263, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987.
Further reading
É. Ghys and P. de la Harpe (editors), Sur les groupes hyperboliques d'après Mikhael Gromov. Progress in Mathematics, 83. Birkhäuser Boston, Inc., Boston, MA, 1990. xii+285 pp. ISBN 0-8176-3508-4