Editing Real tree (section)

=== 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>).