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 (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

${\displaystyle (y,z)_{x}={\frac {1}{2}}{\big (}d(x,y)+d(x,z)-d(y,z){\big )}.}$

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 ${\displaystyle d(x,y)=a+b,\ d(x,z)=a+c,\ d(y,z)=b+c}$. Then the Gromov products are ${\displaystyle (y,z)_{x}=a,\ (x,z)_{y}=b,\ (x,y)_{z}=c}$. 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 the hyperbolic, spherical or euclidean plane, the Gromov product (A, B)_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 c = (a – p) + (b – p), so that p = (a + b – c)/2 = (A,B)_C. Thus for any metric space, a geometric interpretation of (A, B)_C is obtained by isometrically embedding (A, B, C) into the euclidean plane.^[1]

Properties

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

${\displaystyle d(x,y)=(x,z)_{y}+(y,z)_{x},}$

${\displaystyle 0\leq (y,z)_{x}\leq \min {\big \{}d(y,x),d(z,x){\big \}},}$

${\displaystyle {\big |}(y,z)_{p}-(y,z)_{q}{\big |}\leq d(p,q),}$

${\displaystyle {\big |}(x,y)_{p}-(x,z)_{p}{\big |}\leq d(y,z).}$

Points at infinity

Consider hyperbolic space Hⁿ. Fix a base point p and let ${\displaystyle x_{\infty }}$ and ${\displaystyle y_{\infty }}$ be two distinct points at infinity. Then the limit

${\displaystyle \liminf _{x\to x_{\infty } \atop y\to y_{\infty }}(x,y)_{p}}$

exists and is finite, and therefore can be considered as a generalized Gromov product. It is actually given by the formula

${\displaystyle (x_{\infty },y_{\infty })_{p}=\log \csc(\theta /2),}$

where ${\displaystyle \theta }$ is the angle between the geodesic rays ${\displaystyle px_{\infty }}$ and ${\displaystyle py_{\infty }}$.^[2]

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,

${\displaystyle (x,z)_{p}\geq \min {\big \{}(x,y)_{p},(y,z)_{p}{\big \}}-\delta .}$

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).