Δ-hyperbolic space

Δ-hyperbolic space

In mathematics, a "δ"-hyperbolic space is a geodesic metric space in which every geodesic triangle is "δ-thin".

There are many equivalent definitions of "δ"-thin". A simple definition is as follows: pick three points and draw geodesic lines between them to make a geodesic triangle. Then any point on any of the edges of the triangle is within a distance of "δ" from one of the other two sides.

For example, trees are 0-hyperbolic: a geodesic triangle in a tree is just a subtree, so any point on a geodesic triangle is actually on two edges. Normal Euclidean space is ∞-hyperbolic; i.e. not hyperbolic. Generally, the higher "δ" has to be, the less curved the space is.

The definition of "δ"-hyperbolic space is generally credited to Eliyahu Rips. There is also a definition of "δ"-hyperbolicity due to Mikhail Gromov. A geodesic metric space is said to be a Gromov "δ"-hyperbolic space if, for all "p", "x", "y" and "z" in "X",

:(x, z)_{p} geq min ig{ (x, y)_{p}, (y, z)_{p} ig} - delta,

where ("x", "y")"p" denotes the Gromov product of "x" and "y" at "p". "X" is said to be simply Gromov hyperbolic if it is Gromov "δ"-hyperbolic for some "δ" ≥ 0.

ee also

*Negatively curved group

References

* 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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