Real tree

In mathematics, real trees (also called formula_1-trees) are a class of metric spaces generalising simplicial 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 spaces.
Definition and examples.
Formal definition.
A metric space formula_2 is a real tree if it is a geodesic space where every triangle is a tripod. That is, for every three points formula_3 there exists a point formula_4 such that the geodesic segments formula_5 intersect in the segment formula_6 and also formula_7. This definition is equivalent to formula_2 being a "zero-hyperbolic space" in the sense of Gromov (all triangles are "zero-thin"). 
Real trees can also be characterised by a topological property. A metric space formula_2 is a real tree if for any pair of points formula_10 all topological embeddings formula_11 of the segment formula_12 into formula_2 such that formula_14 have the same image (which is then a geodesic segment from formula_15 to formula_16).
In mathematical context.
Real trees often appear, in various situations, as limits of more classical metric spaces.
Brownian trees.
A Brownian tree is a (non-simplicial) real tree almost surely. Brownian trees arise as limits of various random processes on finite trees.
Ultralimits of metric spaces.
Any ultralimit of a sequence formula_25 of formula_26-hyperbolic spaces with formula_27 is a real tree. In particular, the asymptotic cone of any hyperbolic space is a real tree.
Limit of group actions.
Let formula_28 be a group. For a sequence of based formula_28-spaces formula_30 there is a notion of convergence to a based formula_28-space formula_32 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.
A simple example is obtained by taking formula_33 where formula_34 is a compact surface, and formula_35 the universal cover of formula_34 with the metric formula_37 (where formula_38 is a fixed hyperbolic metric on formula_34).
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 properly discontinuously on a real hyperbolic space (this predates Rips', Bestvina's and Paulin's work and is due to J. Morgan and P. Shalen).
Algebraic groups.
If formula_40 is a field with an ultrametric valuation then the Bruhat–Tits building of formula_41 is a real tree. It is simplicial if and only if the valuations is discrete.
Generalisations.
formula_42-trees.
If formula_42 is a totally ordered abelian group there is a natural notion of a distance with values in formula_42 (classical metric spaces correspond to formula_45). There is a notion of formula_42-tree which recovers simplicial trees when formula_47 and real trees when formula_45. The structure of finitely presented groups acting freely on formula_42-trees was described. In particular, such a group acts freely on some formula_50-tree.
Real buildings.
The axioms for a 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.