Gromov product

Gromov product

In mathematics, the Gromov product is a concept in the theory of metric spaces named after the Russian mathematician Mikhail Gromov. Intuitively, the Gromov product measures the distance for which two geodesics starting at the same point remain "close together". 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} = frac1{2} ig( d(x, y) + d(x, z) - d(y, z) ig).

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 ig{ d(y, x), d(z, x) ig},::ig| (y, z)_{p} + (y, z)_{q} ig| leq d(p, q),::ig| (x, y)_{p} + (x, z)_{p} ig| leq d(y, z).

* As mentioned in the introduction, the 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 2"δ" apart (in the sense of the Hausdorff distance between closed sets).

* In fact, 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 ig{ (x, y)_{p}, (y, z)_{p} ig} - delta.

References

* cite book
last = Kapovich
first = Ilya
coauthors = 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 MathSciNet|id=1921706

* cite web
last = Väisälä
first = Jussi
title = Gromov hyperbolic spaces
url = http://www.helsinki.fi/~jvaisala/grobok.pdf
format = PDF
year = 2004
accessdate = 2007-08-28
language = English


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