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

$(y, z)_{x} = \frac1{2} \big( d(x, y) + d(x, z) - d(y, z) \big).$

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 $d(x,y) = a + b, \ d(x,z) = a + c, \ d(y,z) = b + c$. 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.

A tripod with edges of lengths a, b, c. The distance form node x to node y is a+b. The Gromov product (y, z)_x equals a.

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

• 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,

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

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

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

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

Points at infinity

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

$\lim_{x \to x_\infty \atop y \to y_\infty} (x,y)_p$

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

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

where $\theta$ is the angle between the geodesic rays $px_\infty$ and $py_\infty$.^[1]

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,

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

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).

References

1. Roe, John (2003). Lectures on coarse geometry. Providence: American Mathematical Society. p. 114. ISBN 0-8218-3332-4.

• Kapovich, Ilya; Benakli, Nadia (2002). "Boundaries of hyperbolic groups". Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001). Contemp. Math. 296. Providence, RI: Amer. Math. Soc. pp. 39–93. MR 1921706.
• Väisälä, Jussi (2004). "Gromov hyperbolic spaces" (PDF). Retrieved 2007-08-28.