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In [[mathematics]], a '''real tree''', or an <math>\mathbb R</math>-'''tree''', is a [[metric space]] (''M'',''d'') such that
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for any ''x'', ''y'' in ''M'' there is a unique ''arc'' from ''x'' to ''y'' and this arc is a [[Geodesic#Metric_geometry|geodesic segment]]. Here by an ''arc'' from ''x'' to ''y'' we mean the image in ''M'' of a [[homeomorphism|topological embedding]] ''f'' from an [[Interval (mathematics)|interval]] [''a'',''b''] to ''M'' such that ''f''(''a'')=''x'' and ''f''(''b'')=''y''. The condition that the arc is a geodesic segment means that the map ''f'' above can be chosen to be an [[isometry|isometric]] [[embedding]], that is it can be chosen so that for every ''z, t'' in [''a'',''b''] we have ''d(f(z), f(t))''=|''z-t''| and that ''f(a)=x'', ''f(b)=y''.
 
Equivalently, a [[geodesic metric space]] ''M'' is a real tree if and only if ''M'' is a [[δ-hyperbolic space]] with δ=0.
Complete real trees are [[injective metric space]]s {{harv|Kirk|1998}}.
 
There is a theory of [[group action]]s on '''R'''-trees, known as the [[Rips machine]],  which is part of [[geometric group theory]].
 
== Simplicial R-trees ==
A '''simplicial''' '''R'''-tree is an '''R'''-tree that is free from certain "topological strangeness". More precisely, a point ''x'' in an '''R'''-tree ''T'' is called '''ordinary''' if ''T''&minus;''x'' has exactly two components. The points which are not ordinary are '''singular'''.  We define a simplicial '''R'''-tree to be an '''R'''-tree whose set of singular points is [[Discrete space|discrete]] and [[Closed set|closed]].
 
== Examples ==
* Each discrete [[Tree (graph theory)|tree]] can be regarded as an '''R'''-tree by a simple construction such that neighboring vertices have distance one.
* The [[Paris metric]] makes the plane into an '''R'''-tree. If two points are on the same ray in the plane, their distance is defined as the [[Euclidean distance]]. Otherwise, their distance is defined to be the sum of the Euclidian distances of these two points to the origin. More generally any [[hedgehog space]] is an example of a real tree.
* The '''R'''-tree obtained in the following way is nonsimplicial. Start with the interval [0,2] and [[Quotient space#Examples|glue]], for each positive integer ''n'', an interval of length 1/''n'' to the point 1&minus;1/''n'' in the original interval.  The set of singular points is discrete, but fails to be closed since 1 is an ordinary point in this '''R'''-tree. Gluing an interval to 1 would result in a [[closed set]] of singular points at the expense of discreteness.
 
==See also==
*[[Dendroid (topology)]]
*[[Tree-graded space]]
 
== References ==
 
*{{citation
| last = Bestvina | first = Mladen | authorlink = Mladen Bestvina
| contribution = ℝ-trees in topology, geometry, and group theory
| location = Amsterdam
| mr = 1886668
| pages = 55–91
| publisher = North-Holland
| title = Handbook of geometric topology
| url = http://www.math.utah.edu/~bestvina/eprints/handbook.ps
| year = 2002}}.
*{{citation
| last = Chiswell | first = Ian
| isbn = 981-02-4386-3
| location = River Edge, NJ
| mr = 1851337
| publisher = World Scientific Publishing Co. Inc.
| title = Introduction to Λ-trees
| year = 2001}}.
*{{citation
| last = Kirk | first = W. A.
| issue = 1
| journal = Fundamenta Mathematicae
| mr = 1610559
| pages = 67–72
| title = Hyperconvexity of '''R'''-trees
| volume = 156
| url = http://matwbn.icm.edu.pl/ksiazki/fm/fm156/fm15613.pdf
| year = 1998}}.
*{{citation
| last = Shalen | first = Peter B.
| editor-last = Gersten | editor-first = S. M.
| contribution = Dendrology of groups: an introduction
| isbn = 978-0-387-96618-2
| mr = 919830
| pages = 265–319
| publisher = [[Springer-Verlag]]
| series = Math. Sci. Res. Inst. Publ.
| title = Essays in group theory
| volume = 8
| year = 1987}}.
 
[[Category:Group theory]]
[[Category:Geometry]]
[[Category:Topology]]
[[Category:Trees (topology)]]

Latest revision as of 17:57, 11 January 2015

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