Gromov product

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

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

File: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 $d(x,y)\; =\; a\; +\; b,\; \backslash \; d(x,z)\; =\; a\; +\; c,\; \backslash \; d(y,z)\; =\; b\; +\; c$. Then the Gromov products are $(y,z)\_x\; =\; a,\; \backslash \; (x,z)\_y\; =\; b,\; \backslash \; (x,y)\_z\; =\; c$. 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 {{math|1= c = (a – p) + (b – p)}}, so that {{math|1=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.{{Cite journal|date=2005-09-15|title=Gromov hyperbolic spaces|journal=Expositiones Mathematicae|language=en|volume=23|issue=3|pages=187–231|doi=10.1016/j.exmath.2005.01.010|issn=0723-0869|last1=Väisälä|first1=Jussi|doi-access=}}

• 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,

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

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

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

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

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

::$\backslash liminf\_\{x\; \backslash to\; x\_\backslash infty\; \backslash atop\; y\; \backslash to\; y\_\backslash infty\}\; (x,y)\_p$

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

::$(x\_\backslash infty,\; y\_\backslash infty)\_\{p\}\; =\; \backslash log\; \backslash csc\; (\backslash theta/2),$

where $\backslash theta$ is the angle between the geodesic rays $px\_\backslash infty$ and $py\_\backslash infty$.{{cite book|last1=Roe|first1=John|title=Lectures on coarse geometry|date=2003|publisher=American Mathematical Society|location=Providence|isbn=0-8218-3332-4|page=114}}

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,

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

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).

{{reflist}}

• {{citation|language=fr|last1=Coornaert|first=M.|last2=Delzant|first2=T.|last3= Papadopoulos|first3= A.|title=Géométrie et théorie des groupes. Les groupes hyperboliques de Gromov|series=Lecture Notes in Mathematics|volume= 1441|publisher=Springer-Verlag|year= 1990|isbn=3-540-52977-2}}
• {{cite book

| last = Kapovich

| first = Ilya

|author2=Benakli, Nadia

| chapter = Boundaries of hyperbolic groups

| title = Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001)

| series = Contemp. Math. 296

| pages = 39–93

| publisher = Amer. Math. Soc.

| location = Providence, RI

| year = 2002

|mr=1921706

}}

• {{cite journal|last=Väisälä|first=Jussi|title=Gromov hyperbolic spaces|journal=Expositiones Mathematicae|volume=23|issue=3|pages=187–231|doi=10.1016/j.exmath.2005.01.010|year=2005|doi-access=}}

{{Metric spaces}}

Category:Hyperbolic metric space

Category:Metric geometry