Gromov product

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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[edit]

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

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

Motivation[edit]

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 (yz)x equals a.

In Euclidean space, the Gromov product (yz)x equals the distance between x and the point where the inscribed circle of the triangle xyz touches the edge xy.

Properties[edit]

  • The Gromov product is symmetric: (yz)x = (zy)x.
  • The Gromov product degenerates at the endpoints: (yz)y = (yz)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[edit]

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[edit]

The Gromov product can be used to define δ-hyperbolic spaces in the sense of Gromov.: (Xd) 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 (yz)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[edit]

  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.