# real tree

A metric space $X$ is said to be a real tree or $\mathbb{R}$-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 $\mathbb{R}$-tree Related topic MetricSpace Related topic Arc Related topic Curve Related topic SNCFMetric Related topic Isometry Related topic FreeGroup Related topic HyperbolicGroup