PlanetMath: real tree

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real tree

(Definition)

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

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 to the line segment under a (surjective) 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.

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"real tree" is owned by GrafZahl. [ full author list (2) ]

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Other names:

$\mathbb{R}$ -tree

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Cross-references: hyperbolic group, locally finite, surjective, line segment, edges, tree, graph, free group, Cayley graph, hyperbolic metric space, isometric, arc, metric space
There are 3 references to this entry.

This is version 7 of real tree, born on 2005-05-18, modified 2007-06-06.
Object id is 7071, canonical name is RealTree.
Accessed 1936 times total.

Classification:

AMS MSC:	54E40 (General topology :: Spaces with richer structures :: Special maps on metric spaces)
	54E99 (General topology :: Spaces with richer structures :: Miscellaneous)

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