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