A metric space 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 and also. This definition is equivalent to being a "zero-hyperbolic space" in the sense of Gromov. Real trees can also be characterised by a topological property. A metric space is a real tree if for any pair of points all embeddings of the segment into such that have the same image.
Simple examples
If is a graph with the combinatorial metric then it is a real tree if and only if it is a tree. Such a tree is often called a simplicial tree. They are characterised by the following topological property: a real tree is simplicial if and only if the set of singular points of is discrete in.
The R-tree obtained in the following way is nonsimplicial. Start with the interval and glue, for each positive integer n, an interval of length 1/n to the point 1−1/n in the original interval. The set of singular points is discrete, but fails to be closed since 1 is an ordinary point in this R-tree. Gluing an interval to 1 would result in a closed set of singular points at the expense of discreteness.
The Paris metric makes the plane into a real tree. It is defined as follows: one fixes an origin, and if two points are on the same ray from, their distance is defined as the Euclidean distance. Otherwise, their distance is defined to be the sum of the Euclidean distances of these two points to the origin.
More generally any hedgehog space is an example of a real tree.
In mathematical context
Real trees often appear, in various situations, as limits of more classical metric spaces.
Let be a group. For a sequence of based -spaces there is a notion of convergence to a based -space due to M. Bestvina and F. Paulin. When the spaces are hyperbolic and the actions are unbounded the limit is a real tree. A simple example is obtained by taking where is a compact surface, and the universal cover of with the metric . 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.
If is a totally ordered abelian group there is a natural notion of a distance with values in . There is a notion of -tree which recovers simplicial trees and real trees. The structure of finitely presented groups acting freely on -trees was described. In particular, such a group acts freely on some -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.
