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单词 RealTree
释义

real tree


A metric space X is said to be a real tree orR-tree, if for each x,yX there is a unique arcfrom x to y, and furthermore this arc is an isometric http://planetmath.org/node/429embeddingMathworldPlanetmathPlanetmathPlanetmath.

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 (http://planetmath.org/Isometric) to the line segment [0,1] under a (surjectivePlanetmathPlanetmath) isometry (http://planetmath.org/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.

Titlereal tree
Canonical nameRealTree
Date of creation2013-03-22 15:16:55
Last modified on2013-03-22 15:16:55
OwnerGrafZahl (9234)
Last modified byGrafZahl (9234)
Numerical id10
AuthorGrafZahl (9234)
Entry typeDefinition
Classificationmsc 54E99
Classificationmsc 54E40
Synonym-tree
Related topicMetricSpace
Related topicArc
Related topicCurve
Related topicSNCFMetric
Related topicIsometry
Related topicFreeGroup
Related topicHyperbolicGroup
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