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

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

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.

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.

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

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

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

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

where $\theta$ is the angle between the geodesic rays $px_{\infty }$ and $py_{\infty }$.

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,

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

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

• Coornaert, M.; Delzant, T.; Papadopoulos, A. (1990), Géométrie et théorie des groupes. Les groupes hyperboliques de Gromov, Lecture Notes in Mathematics (in French), vol. 1441, Springer-Verlag, ISBN 3-540-52977-2
• 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 (2005). "Gromov hyperbolic spaces". Expositiones Mathematicae. 23 (3): 187–231. doi:10.1016/j.exmath.2005.01.010.