real treeRealTree2005-05-18 11:18:462007-06-06 01:11:31Definition% this is the default PlanetMath preamble. as your knowledge
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\DeclareMathOperator{\wt}{wt}A metric space $X$ is said to be a \emph{real tree} or
\emph{$\mbb{R}$-tree}, if for each $x,y\in X$ there is a unique arc
from $x$ to $y$, and furthermore this arc is an isometric \PMlinkid{embedding}{429}.
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 \PMlinkname{isometric}{Isometric} to the line segment $[0,1]$ under a (surjective) \PMlinkname{isometry}{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.