Real tree

 ( Group Theory ) 

In mathematics, real trees (also called $\backslash mathbb\; R$-trees) are a class of 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.

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Definition and examples

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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 $x,\; y,\; \backslash rho\; \backslash in\; X$ there exists a point $c\; =\; x\; \backslash wedge\; y$ such that the geodesic segments $\backslash rho,x,\; \backslash rho,y$ intersect in the segment $\backslash rho,c$ and also $c\; \backslash in\; x,y$. 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 topology property. A metric space $X$ is a real tree if for any pair of points $x,\; y\; \backslash in\; X$ all topological embeddings $\backslash sigma$ of the segment $0,1$ into $X$ such that $\backslash sigma(0)\; =\; x,\; \backslash ,\; \backslash sigma(1)\; =\; y$ have the same image (which is then a geodesic segment from $x$ to $y$).

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Simple examples

• If $X$ 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 $T$ is simplicial if and only if the set of singular points of $X$ (points whose complement in $X$ has three or more connected components) is closed and discrete in $X$.
• The $\backslash mathbb\; R$-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 $\backslash mathbb\; R$-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 $P$, and if two points are on the same ray from $P$, 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 $P$.
• 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.

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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 Homeomorphism to a circle.

2) A real tree is a connected metric space $(X,d)$ which has the four points conditionPeter Buneman, A Note on the Metric Properties of Trees, Journal of combinatorial theory, B (17), , 1974. (see figure):

For all $x,y,z,t\backslash in\; X,$ $d(x,y)+d(z,t)\backslash leq\; \backslash maxd(x,z)+d(y,t)\backslash ,;\backslash ,$.

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

For all $x,y,z,t\backslash in\; X,$ $(x,y)\_t\backslash geq\; \backslash min\; ,$

where $(x,y)\_t$ denotes the Gromov product of $x$ and $y$ with respect to $t$, that is, $\backslash textstyle\backslash frac\; 1\; 2\; \backslash left(\; d(x,\; t)\; +\; d(y,\; t)\; -\; d(x,\; y)\; \backslash right).$

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 $t\backslash in\; ]a,b[$.

For $x,\; y\backslash in\; a,b$, $x\backslash leq\; y$, define a Metric space and an equivalence relation with:

$d\_e(\; x,\; y)\; :=\; e(x)+e(y)-2\backslash min(e(z)\backslash ,\; ;z\backslash inx,y),$

$x\backslash sim\_e\; y\; \backslash Leftrightarrow\; d\_e(x,y)=0.$

Then, the quotient space $(a,b/\backslash sim\_e\backslash ,\; ,\backslash ,\; d\_e)$ 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).

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

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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.

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Ultralimits of metric spaces Any ultralimit of a sequence $(X\_i)$ of $\backslash delta\_i$-hyperbolic spaces with $\backslash delta\_i\; \backslash to\; 0$ is a real tree. In particular, the asymptotic cone of any hyperbolic space is a real tree.

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Limit of group actions Let $G$ be a group. For a sequence of based $G$-spaces $(X\_i,\; *\_i,\; \backslash rho\_i)$ there is a notion of convergence to a based $G$-space $(X\_\backslash infty,\; x\_\backslash infty,\; \backslash rho\_\backslash infty)$ 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 $G\; =\; \backslash pi\_1(S)$ where $S$ is a compact space surface, and $X\_i$ the universal cover of $S$ with the metric $i\backslash rho$ (where $\backslash rho$ 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 Hyperbolic space (this predates Rips', Bestvina's and Paulin's work and is due to J. Morgan and Peter Shalen).

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Algebraic groups If $F$ is a field with an ultrametric valuation then the Bruhat–Tits building of $\backslash mathrm\{SL\}\_2(F)$ is a real tree. It is simplicial if and only if the valuations is discrete.

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Generalisations

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

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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.

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See also

• Dendroid (topology)
• Tree-graded space

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