In mathematics, real trees (also called \mathbb R-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 X is a real tree if it is a geodesic space where every triangle is a tripod. That is, for every three points there exists a point such that the geodesic segments intersect in the segment [\rho,c] and also c \in [x,y]. This definition is equivalent to X 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 X is a real tree if for any pair of points x, y \in X all topological embeddings \sigma of the segment [0,1] into X such that have the same image (which is then a geodesic segment from x to y).

Simple examples

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.
2. A real tree is a connected metric space (X,d) which has the four points condition (see figure):
3. A real tree is a connected 0-hyperbolic metric space (see figure). Formally, where (x,y)_t denotes the Gromov product of x and y with respect to t, that is,
4. (similar to the characterization of plane trees by their contour process). Consider a positive excursion of a function. In other words, let e be a continuous real-valued function and [a,b] an interval such that e(a)=e(b)=0 and e(t)>0 for t\in ]a,b[. For, x\leq y, 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 stochastic process 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 of a sequence (X_i) of \delta_i-hyperbolic spaces with is a real tree. In particular, the asymptotic cone of any hyperbolic space is a real tree.

Limit of group actions

Let G be a group. For a sequence of based G-spaces there is a notion of convergence to a based G-space 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 where S is a compact surface, and X_i the universal cover of S with the metric i\rho (where \rho is a fixed hyperbolic metric on S). 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 F is a field with an ultrametric valuation then the Bruhat–Tits building of is a real tree. It is simplicial if and only if the valuations is discrete.

Generalisations

\Lambda -trees

If \Lambda is a totally ordered abelian group there is a natural notion of a distance with values in \Lambda (classical metric spaces correspond to ). There is a notion of \Lambda-tree which recovers simplicial trees when and real trees when. The structure of finitely presented groups acting freely on \Lambda-trees was described. In particular, such a group acts freely on some \mathbb R^n-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.