In mathematics, the Gromov product is a concept in the theory of metric spaces named after the mathematician Mikhail Gromov. The Gromov product can also be used to define δ-hyperbolic metric spaces in the sense of Gromov.
Let (X, d) be a metric space and let x, y, z â X. Then the Gromov product of y and z at x, denoted (y, z)<sub>x</sub>, is defined by
Given three points x, y, z in the metric space X, by the triangle inequality there exist non-negative numbers a, b, c such that . Then the Gromov products are . In the case that the points x, y, z are the outer nodes of a tripod then these Gromov products are the lengths of the edges.
In the hyperbolic, spherical or euclidean plane, the Gromov product (A, B)<sub>C</sub> equals the distance p between C and the point where the incircle of the geodesic triangle ABC touches the edge CB or CA. Indeed from the diagram , so that . Thus for any metric space, a geometric interpretation of (A, B)<sub>C</sub> is obtained by isometrically embedding (A, B, C) into the euclidean plane.
Consider hyperbolic space H<sup>n</sup>. Fix a base point p and let and be two distinct points at infinity. Then the limit
exists and is finite, and therefore can be considered as a generalized Gromov product. It is actually given by the formula
where is the angle between the geodesic rays and .
The Gromov product can be used to define δ-hyperbolic spaces in the sense of Gromov.: (X, d) is said to be ô-hyperbolic if, for all p, x, y and z in X,
In this case. Gromov product measures how long geodesics remain close together. Namely, if x, y and z are three points of a ô-hyperbolic metric space then the initial segments of length (y, z)<sub>x</sub> of geodesics from x to y and x to z are no further than 2ô apart (in the sense of the Hausdorff distance between closed sets).