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

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

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 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 Template:Math, so that Template:Math. Thus for any metric space, a geometric interpretation of (A, B)_C is obtained by isometrically embedding (A, B, C) into the euclidean plane.

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,

$d(x,y)=(x,z{)}_y+(y,z{)}_x,$

$0\le (y,z{)}_x\le min\{d(y,x),d(z,x)\},$

$|(y,z{)}_p-(y,z{)}_q|\le d(p,q),$

$|(x,y{)}_p-(x,z{)}_p|\le d(y,z).$

Points at infinity

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

$\underset{\genfrac{}{}{0ex}{}{x\to x_{\mathrm{\infty }}}{y\to y_{\mathrm{\infty }}}}{lim\; inf}(x,y{)}_p$

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

$(x_{\mathrm{\infty }},y_{\mathrm{\infty }}{)}_p=\log \csc (\theta /2),$

where $\theta$ is the angle between the geodesic rays $p{x}_{\mathrm{\infty }}$ and $p{y}_{\mathrm{\infty }}$.

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,

$(x,z{)}_p\ge min\{(x,y{)}_p,(y,z{)}_p\}-\delta .$

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).

References

• Template:Citation
• Template:Cite book
• Template:Cite journal