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

Motivation[edit]

Inkreis mit Strecken.svg

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 . Then the Gromov products are . 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 (AB)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 = (ap) + (bp), so that p = (a + bc)/2 = (A,B)C

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,

Points at infinity[edit]

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

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

where is the angle between the geodesic rays and .[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,

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).

Notes[edit]

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

References[edit]

  • Coornaert, M.; Delzant, T.; Papadopoulos, A. (1990), Géométrie et théorie des groupes. Les groupes hyperboliques de Gromov, Lecture Notes in Mathematics (in French), 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 (2004). "Gromov hyperbolic spaces" (PDF). Retrieved 2007-08-28.