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Viewing Version
6
of
'real tree'
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| Title of object: |
real tree |
| Canonical Name: |
RealTree |
| Type: |
Definition |
| Created on: |
2005-05-18 11:18:46 |
| Modified on: |
2007-06-06 01:09:02 |
| Classification: |
msc:54E40, msc:54E99 |
| Synonyms: |
real tree=$\mathbb{R}$-tree |
Revision comment (for changes between this and next version):
removed "isometric" from related section (redundant due to "isometry")
added "free group" and "hyperbolic group" to related section |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage[latin1]{inputenc}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\newcommand{\Bigcup}{\bigcup\limits}
\newcommand{\Prod}{\prod\limits}
\newcommand{\Sum}{\sum\limits}
\newcommand{\mbb}{\mathbb}
\newcommand{\mbf}{\mathbf}
\newcommand{\mc}{\mathcal}
\newcommand{\mmm}[9]{\left(\begin{array}{rrr}#1\\#4\\#7	\end{array}\right)}
\newcommand{\ol}{\overline}
% Math Operators/functions
\DeclareMathOperator{\Frob}{Frob}
\DeclareMathOperator{\cwe}{cwe}
\DeclareMathOperator{\we}{we}
\DeclareMathOperator{\wt}{wt} |
Content:
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. |
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