real tree

A metric space $X$ is said to be a real tree or $\mathrm{R}$-tree, if for each $x,y\in X$ there is a unique arc from $x$ to $y$, and furthermore this arc is an isometric http://planetmath.org/node/429embedding.

Every real tree is a hyperbolic metric space; moreover, every real tree is 0 hyperbolic.

The Cayley graph of any free group is considered to be a real tree. Note that its graph is a tree in the graph theoretic sense. To make it a real tree, we view the edges as isometric (http://planetmath.org/Isometric) to the line segment $[0,1]$ under a (surjective) isometry (http://planetmath.org/Isometry) and attach the edges to the tree. The resulting 1-complex is then a locally finite real tree. Because of this result, every free group is a hyperbolic group.

Title	real tree
Canonical name	RealTree
Date of creation	2013-03-22 15:16:55
Last modified on	2013-03-22 15:16:55
Owner	GrafZahl (9234)
Last modified by	GrafZahl (9234)
Numerical id	10
Author	GrafZahl (9234)
Entry type	Definition
Classification	msc 54E99
Classification	msc 54E40
Synonym	$ℝ$-tree
Related topic	MetricSpace
Related topic	Arc
Related topic	Curve
Related topic	SNCFMetric
Related topic	Isometry
Related topic	FreeGroup
Related topic	HyperbolicGroup