| Version 4 |
Version 3 |
| A metric space $X$ is said to be a \emph{real tree} or |
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 |
\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}. |
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. |
Every real tree is a hyperbolic metric space; moreover, every real tree is 0 hyperbolic. |
|
|
|
The Cayley graph of any free group is a real tree: it is a tree in the graph theoretic sence; to make it a real tree, we view the edges as isomorphic to the $[0,1]$ segment and attach them to the tree; the resulting 1-complex is then a locally finite real tree. Thus, every free group is a hyperbolic group.
|
The Cayley graph of any free group is a real tree. Thus, every free group is a hyperbolic group.
|
