Gromov product


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.

Definition

Let be a metric space and let x, y, z ∈ X. Then the Gromov product of y and z at x, denoted x, is defined by

Motivation

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 C 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 C is obtained by isometrically embedding into the euclidean plane.

Properties

  • The Gromov product is symmetric: x = x.
  • The Gromov product degenerates at the endpoints: y = z = 0.
  • For any points p, q, x, y and z,

Points at infinity

Consider hyperbolic space Hn. 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.

δ-hyperbolic spaces and divergence of geodesics

The Gromov product can be used to define δ-hyperbolic spaces in the sense of Gromov.: is said to be δ-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 δ-hyperbolic metric space then the initial segments of length x of geodesics from x to y and x to z are no further than 2δ apart.