Editing Real tree

In [[mathematics]], '''real trees''' (also called '''<math>\mathbb R</math>-trees''') are a class of [[metric space]]s generalising simplicial [[Tree (graph theory)|trees]]. They arise naturally in many mathematical contexts, in particular [[geometric group theory]] and [[probability theory]]. They are also the simplest examples of [[Gromov hyperbolic space]]s.

== Definition and examples ==

=== Formal definition ===

[[File:Y property.png|130px|thumb|A triangle in a real tree]]

A metric space <math>X</math> is a real tree if it is a [[Geodesic metric space|geodesic space]] where every triangle is a tripod. That is, for every three points <math>x, y, \rho \in X</math> there exists a point <math>c = x \wedge y</math> such that the geodesic segments <math>[\rho,x], [\rho,y]</math> intersect in the segment <math>[\rho,c]</math> and also <math>c \in [x,y]</math>. This definition is equivalent to <math>X</math> being a "zero-hyperbolic space" in the sense of Gromov (all triangles are "zero-thin"). 
Real trees can also be characterised by a [[topology|topological]] property. A metric space <math>X</math> is a real tree if for any pair of points <math>x, y \in X</math> all [[topological embedding]]s <math>\sigma</math> of the segment <math>[0,1]</math> into <math>X</math> such that <math>\sigma(0) = x, \, \sigma(1) = y</math> have the same image (which is then a geodesic segment from <math>x</math> to <math>y</math>).

=== Simple examples ===

*If <math>X</math> is a connected graph with the combinatorial metric then it is a real tree if and only if it is a tree (i.e. it has no [[Cycle (graph theory)|cycles]]). Such a tree is often called a simplicial tree. They are characterised by the following topological property: a real tree <math>T</math> is simplicial if and only if the set of singular points of <math>X</math> (points whose complement in <math>X</math> has three or more connected components) is closed and discrete in <math>X</math>.
* The <math>\mathbb R</math>-tree obtained in the following way is nonsimplicial. Start with the interval [0,&thinsp;2] and glue, for each positive integer ''n'', an interval of length 1/''n'' to the point 1&thinsp;−&thinsp;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 <math>\mathbb R</math>-tree. Gluing an interval to 1 would result in a [[closed set]] of singular points at the expense of discreteness.
* The [[Paris metric]] makes the plane into a real tree. It is defined as follows: one fixes an origin <math>P</math>, and if two points are on the same ray from <math>P</math>, their distance is defined as the Euclidean distance. Otherwise, their distance is defined to be the sum of the Euclidean distances of these two points to the origin <math>P</math>.
* The plane under the Paris metric is an example of a [[hedgehog space]], a collection of line segments joined at a common endpoint. Any such space is a real tree.

== Characterizations ==
[[File:Four_point_condition.png|thumb|260x260px|Visualisation of the four points condition and the 0-hyperbolicity. In green: <math>(x,y)_t=(y,z)_t</math> ; in blue: <math>(x,z)_t</math>.]]
Here are equivalent characterizations of real trees which can be used as definitions:

1) ''(similar to [[Tree (data structure)|trees]] as graphs)'' A real tree is a [[Intrinsic_metric|geodesic]] [[metric space]] which contains no subset [[Homeomorphism|homeomorphic]] to a circle.<ref>{{Cite book |last=Chiswell |first=Ian |url=https://www.worldcat.org/oclc/268962256 |title=Introduction to [lambda]-trees |date=2001 |publisher=World Scientific |isbn=978-981-281-053-3 |location=Singapore |oclc=268962256}}</ref>

2) A real tree is a connected metric space <math>(X,d)</math> which has the '''four points condition'''<ref>Peter Buneman, ''A Note on the Metric Properties of Trees'', Journal of combinatorial theory, B (17), {{p.|48-50}}, 1974.</ref> (see figure): 
:For all <math>x,y,z,t\in X,</math>  <math> d(x,y)+d(z,t)\leq \max[d(x,z)+d(y,t)\,;\, d(x,t)+d(y,z)]</math>.

3) A real tree is a connected [[Δ-hyperbolic space|0-hyperbolic]] metric space<ref name=":0">{{Cite book |last=Evans |first=Stevan N. |title=Probability and Real Trees |publisher=École d’Eté de Probabilités de Saint-Flour XXXV |year=2005}}</ref> (see figure). Formally, 
:For all <math>x,y,z,t\in X,</math>  <math> (x,y)_t\geq \min [ (x,z)_t\, ; \, (y,z)_t ],</math>
where <math>(x,y)_t</math> denotes the [[Gromov product]] of <math>x</math> and <math>y</math> with respect to <math>t</math>, that is, <math>\textstyle\frac 1 2 \left( d(x, t) + d(y, t) - d(x, y) \right).</math>

4) ''(similar to the characterization of [[Tree (graph theory)#Plane tree|plane trees]] by their [[Galton-Watson tree|contour process]]).'' Consider a positive excursion of a function. In other words, let <math>e</math> be a continuous real-valued function and <math>[a,b]</math> an interval such that <math>e(a)=e(b)=0</math> and <math>e(t)>0</math> for <math>t\in ]a,b[</math>.

For <math>x, y\in [a,b]</math>, <math>x\leq y</math>, define a [[Metric space|pseudometric]] and an [[equivalence relation]] with: 

:<math> d_e( x, y) := e(x)+e(y)-2\min(e(z)\, ;z\in[x,y]), </math> 

:<math> x\sim_e y \Leftrightarrow d_e(x,y)=0.</math>  

Then, the [[Quotient space (topology)|quotient space]] <math>([a,b]/\sim_e\, ,\, d_e) </math> is a real tree.<ref name=":0" />   Intuitively, the [[local minima]] of the excursion ''e'' are the parents of the [[local maxima]]. Another visual way to construct the real tree from an excursion is to "put glue" under the curve of ''e'', and "bend" this curve, identifying the glued points (see animation).

[[File:Collage.ogg|center|thumb|300x300px|Partant d'une [[Excursion brownienne|excursion]] ''e'' (en noir), la déformation (en vert) représente le « pliage » de la courbe jusqu'au « collage » des points d'une même classe d'équivalence, l'état final est l'arbre réel associé à ''e''.]]

== Examples ==

Real trees often appear, in various situations, as limits of more classical metric spaces.

=== Brownian trees ===

A [[Brownian tree]]<ref>{{citation | last = Aldous | first = D. | author-link = David Aldous | date = 1991 | title = The continuum random tree I | journal = [[Annals of Probability]] | volume = 19 | pages = 1–28| doi = 10.1214/aop/1176990534 | doi-access = free }}</ref> is a random metric space whose value is a (non-simplicial) real tree almost surely. Brownian trees arise as limits of various random processes on finite trees.<ref>{{citation | last = Aldous | first = D. | author-link = David Aldous | date = 1991 | title = The continuum random tree III | journal = [[Annals of Probability]] | volume = 21 | pages = 248–289}}</ref>

=== Ultralimits of metric spaces ===

Any [[ultralimit]] of a sequence <math>(X_i)</math> of <math>\delta_i</math>-[[Hyperbolic metric space|hyperbolic]] spaces with <math>\delta_i \to 0</math> is a real tree. In particular, the [[Ultralimit#Asymptotic cones|asymptotic cone]] of any hyperbolic space is a real tree.

=== Limit of group actions ===

Let <math>G</math> be a [[group (mathematics)|group]]. For a sequence of based <math>G</math>-spaces <math>(X_i, *_i, \rho_i)</math> there is a notion of convergence to a based <math>G</math>-space <math>(X_\infty, x_\infty, \rho_\infty)</math> due to M. Bestvina and F. Paulin. When the spaces are hyperbolic and the actions are unbounded the limit (if it exists) is a real tree.<ref>{{citation | last = Bestvina | first = Mladen | author-link = Mladen Bestvina | title = Handbook of Geometric Topology | contribution = <math>\mathbb R</math>-trees in topology, geometry and group theory | pages = 55–91 | year = 2002 | publisher = Elsevier | isbn = 9780080532851 | url = https://books.google.com/books?id=8OYxdADnhZoC&pg=PA55}}</ref>

A simple example is obtained by taking <math>G = \pi_1(S)</math> where <math>S</math> is a [[compact space|compact]] surface, and <math>X_i</math> the universal cover of <math>S</math> with the metric <math>i\rho</math> (where <math>\rho</math> is a fixed hyperbolic metric on <math>S</math>).

This is useful to produce actions of hyperbolic groups on real trees. Such actions are analyzed using the so-called [[Rips machine]]. A case of particular interest is the study of degeneration of groups acting [[Group_action#Remarkable properties of actions|properly discontinuously]] on a [[Hyperbolic space|real hyperbolic space]] (this predates Rips', Bestvina's and Paulin's work and is due to J. Morgan and [[Peter Shalen|P. Shalen]]<ref>{{citation
 | last = Shalen | first = Peter B. | author-link = Peter Shalen
 | 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}}</ref>).

=== Algebraic groups ===

If <math>F</math> is a [[field (mathematics)|field]] with an [[ultrametric space|ultrametric]] [[Valuation (algebra)|valuation]] then the [[Building (mathematics)|Bruhat–Tits building]] of <math>\mathrm{SL}_2(F)</math> is a real tree. It is simplicial if and only if the valuations is discrete.

== Generalisations ==

=== Λ-trees ===

If <math>\Lambda</math> is a [[totally ordered abelian group]] there is a natural notion of a distance with values in <math>\Lambda</math> (classical metric spaces correspond to <math>\Lambda = \mathbb R</math>). There is a notion of <math>\Lambda</math>-tree<ref>{{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}}</ref> which recovers simplicial trees when <math>\Lambda = \mathbb Z</math> and real trees when <math>\Lambda = \mathbb R</math>. The structure of [[finitely presented group]]s acting [[Group_action#Remarkable properties of actions|freely]] on <math>\Lambda</math>-trees was described. <ref>{{citation | last = O. Kharlampovich, A. Myasnikov, D. Serbin | title = Actions, length functions and non-archimedean words IJAC 23, No. 2, 2013.}}</ref> In particular, such a group acts freely on some  <math>\mathbb R^n</math>-tree.

=== Real buildings ===

The axioms for a [[Building (mathematics)|building]] can be generalized to give a definition of a real building. These arise for example as asymptotic cones of higher-rank [[symmetric spaces]] or as Bruhat-Tits buildings of higher-rank groups over valued fields.

==See also==
*[[Dendroid (topology)]]
*[[Tree-graded space]]

== References ==

{{reflist}}

[[Category:Group theory]]
[[Category:Geometry]]
[[Category:Topology]]
[[Category:Trees (topology)]]