Gromov product

Definition

Let (X, d) be a metric space and let x, y, z ∈ X. Then the Gromov product of y and z at x, denoted (y, z)_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 d ( x , y ) = a + b ,   d ( x , z ) = a + c ,   d ( y , z ) = b + c . Then the Gromov products are ( y , z ) x = a ,   ( x , z ) y = b ,   ( x , y ) z = c . 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 Euclidean space, the Gromov product (y, z)_x equals the distance between x and the point where the inscribed circle of the triangle xyz touches the edge xy.

Properties

Points at infinity

Consider hyperbolic space Hⁿ. Fix a base point p and let x ∞ and y ∞ 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 θ is the angle between the geodesic rays p x ∞ and p y ∞ .

δ-hyperbolic spaces and divergence of geodesics

The Gromov product can be used to define δ-hyperbolic spaces in the sense of Gromov.: (X, d) 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 (y, z)_x of geodesics from x to y and x to z are no further than 2δ apart (in the sense of the Hausdorff distance between closed sets).