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. Group theory is that branch of mathematics concerned with the study of groups. ... This picture illustrates how the hours in a clock form a group. ... In mathematics, the word metric is a metric defined on a group, depending on a set of generators for the group. ... Lines through a given point P and hyperparallel to line l. ...

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 finite 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. Eliyahu Rips is an Israeli mathematician known for his research in algebra and the controversial Bible codes. ... The Cayley graph of the free group on two generators a and b In mathematics, a Cayley graph, named after Arthur Cayley, is a graph that encodes the structure of a group. ...

A triangle in T is simply three points (the vertices) with each pair being joined by a geodesic segment (a side). Fix . 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. In mathematics, a -hyperbolic space is a geodesic metric space that satisfies a thin triangles condition. ...

References

Mikhail Gromov, Hyperbolic groups. Essays in group theory, 75--263, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987. Mikhail Leonidovich Gromov Russian: Михаил Леонидович Громов (born December 23, 1943, also known as Mikhael Gromov, Michael Gromov, or Misha Gromov) is a mathematician known for important contributions in many different areas of geometry, especially metric geometry, symplectic geometry, and geometric group theory. ...

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

Michel Coornaert, Thomas Delzant, Athanase Papadopoulos, "Géométrie et théorie des groupes : les groupes hyperboliques de Gromov", Lecture Notes in Mathematics, vol. 1441, Springer-Verlag, Berlin, 1990, MR 92f:57003, ISBN 3-540-52977-2