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.

Definition and examples

Formal definition

is a real tree if it is a geodesic space being a "zero-hyperbolic space" in the sense of Gromov (all triangles are "zero-thin").

Real trees can also be characterised by a topological all topological embeddings).

Simple examples

• is a connected graph with the combinatorial metric then it is a real tree if and only if it is a tree (i.e. it has no cycles.

• -tree obtained in the following way is nonsimplicial. Start with the interval [0,2] and glue, for each positive integer n, an interval of length 1/n to the point 1−1/n-tree. Gluing an interval to 1 would result in a closed set of singular points at the expense of discreteness.

• The Paris metric.

• The plane under the Paris metric is an example of a hedgehog space, a collection of line segments joined at a common endpoint. Any such space is a real tree.

Characterizations

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.

which has the four points condition (see figure):

1. .

3) A real tree is a connected 0-hyperbolic metric space (see figure). Formally,

denotes the Gromov product

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

, define a pseudometric and an equivalence relation with:

Then, the quotient space is a real tree. Intuitively, the local minima of the excursion e are the parents of the local maxima. 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).

Examples

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

Brownian trees

A Brownian tree 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.

Ultralimits of metric spaces

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

Limit of group actions

be a group 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.

is a compact).

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

is a field with an ultrametric valuation then the Bruhat–Tits building is a real tree. It is simplicial if and only if the valuations is discrete.

Generalisations

Λ-trees

is a totally ordered abelian group-tree. The structure of finitely presented groups acting freely-trees was described. -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.

See also

• Dendroid (topology)

• Tree-graded space

References