In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations (depending quantitatively on a nonnegative real number δ) between points. The definition, introduced by Mikhael Gromov, generalizes the metric properties of classical hyperbolic geometry and of trees. Hyperbolicity is a large-scale property, and is very useful to the study of certain infinite groups called Gromov-hyperbolic groups. Definitions In this paragraph we give various definitions of a -hyperbolic space. A metric space is said to be (Gromov-) hyperbolic if it is -hyperbolic for some . Definition using the Gromov product Let be a metric space. The Gromov product of two points with respect to a third one is defined by the formula: Gromov's definition of a hyperbolic metric space is then as follows: is -hyperbolic if and only if all satisfy the four-point condition Note that if this condition is satisfied for all and one fixed base point , then it is satisfied for all with a constant . Thus the hyperbolicity condition only needs to be verified for one fixed base point; for this reason, the subscript for the base point is often dropped from the Gromov product. Definitions using triangles Up to changing by a constant multiple, there is an equivalent geometric definition involving triangles when the metric space is geodesic, i.e. any two points are end points of a geodesic segment (an isometric image of a compact subinterval of the reals). Note that the definition via Gromov products does not require the space to be geodesic. Let . A geodesic triangle with vertices is the union of three geodesic segments (where denotes a segment with endpoints and ). If for any point there is a point in at distance less than of , and similarly for points on the other edges, and then the triangle is said to be -slim . A definition of a -hyperbolic space is then a geodesic metric space all of whose geodesic triangles are -slim. This definition is generally credited to Eliyahu Rips. Another definition can be given using the notion of a -approximate center of a geodesic triangle: this is a point which is at distance at most of any edge of the triangle (an "approximate" version of the incenter). A space is -hyperbolic if every geodesic triangle has a -center. These two definitions of a -hyperbolic space using geodesic triangles are not exactly equivalent, but there exists