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

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Formal definition

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A triangle in a real tree

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 (all triangles are "zero-thin"). Real trees can also be characterised by a topological property. A metric space   is a real tree if for any pair of points   all topological embeddings   of the segment   into   such that   have the same image (which is then a geodesic segment from   to  ).

Simple examples

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  • If   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). 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   (points whose complement in   has three or more connected components) is closed and discrete in  .
  • The  -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 in the original interval. The set of singular points is discrete, but fails to be closed since 1 is an ordinary point in this  -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  .
  • 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

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Visualisation of the four points condition and the 0-hyperbolicity. In green:   ; in blue:  .

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):

For all    .

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

For all    

where   denotes the Gromov product of   and   with respect to  , 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   be a continuous real-valued function and   an interval such that   and   for  .

For  ,  , define a pseudometric and an equivalence relation with:

 
 

Then, the quotient space   is a real tree.[3] 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).

Partant d'une excursion e (en noir), la déformation (en vert) représente le « pliage » de la courbe jusqu'au « collage » des points d'une même classe d'équivalence, l'état final est l'arbre réel associé à e.

Examples

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

Brownian trees

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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.[5]

Ultralimits of metric spaces

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

Limit of group actions

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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 (if it exists) is a real tree.[6]

A simple example is obtained by taking   where   is a compact surface, and   the universal cover of   with the metric   (where   is a fixed hyperbolic metric on  ).

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]).

Algebraic groups

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If   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

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 -trees

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If   is a totally ordered abelian group there is a natural notion of a distance with values in   (classical metric spaces correspond to  ). There is a notion of  -tree[8] which recovers simplicial trees when   and real trees when  . The structure of finitely presented groups acting freely on  -trees was described. [9] In particular, such a group acts freely on some  -tree.

Real buildings

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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

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References

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  1. ^ Chiswell, Ian (2001). Introduction to [lambda]-trees. Singapore: World Scientific. ISBN 978-981-281-053-3. OCLC 268962256.
  2. ^ Peter Buneman, A Note on the Metric Properties of Trees, Journal of combinatorial theory, B (17), p. 48-50, 1974.
  3. ^ a b Evans, Stevan N. (2005). Probability and Real Trees. École d’Eté de Probabilités de Saint-Flour XXXV.
  4. ^ Aldous, D. (1991), "The continuum random tree I", Annals of Probability, 19: 1–28, doi:10.1214/aop/1176990534
  5. ^ Aldous, D. (1991), "The continuum random tree III", Annals of Probability, 21: 248–289
  6. ^ Bestvina, Mladen (2002), " -trees in topology, geometry and group theory", Handbook of Geometric Topology, Elsevier, pp. 55–91, ISBN 9780080532851
  7. ^ Shalen, Peter B. (1987), "Dendrology of groups: an introduction", in Gersten, S. M. (ed.), Essays in Group Theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer-Verlag, pp. 265–319, ISBN 978-0-387-96618-2, MR 0919830
  8. ^ Chiswell, Ian (2001), Introduction to Λ-trees, River Edge, NJ: World Scientific Publishing Co. Inc., ISBN 981-02-4386-3, MR 1851337
  9. ^ O. Kharlampovich, A. Myasnikov, D. Serbin, Actions, length functions and non-archimedean words IJAC 23, No. 2, 2013.{{citation}}: CS1 maint: multiple names: authors list (link)