Editing Real tree (section)

== 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.