Editing Real tree (section)

== 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.