Real tree

In mathematics, real trees (also called

-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.

is a real tree if it is a geodesic space where every triangle is a tripod.

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.

is a real tree if for any pair of points

have the same image (which is then a geodesic segment from

Here are equivalent characterizations of real trees which can be used as definitions: 1) (similar to trees as graphs) A real tree is a geodesic metric space which contains no subset homeomorphic to a circle.

[1] 2) A real tree is a connected metric space

which has the four points condition[2] (see figure): 3) A real tree is a connected 0-hyperbolic metric space[3] (see figure).

4) (similar to the characterization of plane trees by their contour process).

Consider a positive excursion of a function.

, define a pseudometric and an equivalence relation with: Then, the quotient space

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).

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

A Brownian tree[4] is a stochastic process whose value is a (non-simplicial) real tree almost surely.

Brownian trees arise as limits of various random processes on finite trees.

In particular, the asymptotic cone of any hyperbolic space is a real tree.

When the spaces are hyperbolic and the actions are unbounded the limit (if it exists) is a real tree.

[6] A simple example is obtained by taking

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[7]).

is a field with an ultrametric valuation then the Bruhat–Tits building of

is a totally ordered abelian group there is a natural notion of a distance with values in

(classical metric spaces correspond to

The structure of finitely presented groups acting freely on

These arise for example as asymptotic cones of higher-rank symmetric spaces or as Bruhat-Tits buildings of higher-rank groups over valued fields.