real tree

A metric space X is said to be a real tree or R-tree, if for each x,yX there is a unique arc from x to y, and furthermore this arc is an isometric

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

The Cayley graphMathworldPlanetmath of any free groupMathworldPlanetmath 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 ( to the line segment [0,1] under a (surjectivePlanetmathPlanetmath) isometry ( and attach the edges to the tree. The resulting 1-complex is then a locally finitePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath 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 -tree
Related topic MetricSpace
Related topic Arc
Related topic Curve
Related topic SNCFMetric
Related topic Isometry
Related topic FreeGroup
Related topic HyperbolicGroup