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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, real trees (also called \mathbb R-trees) are a class of
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s generalising simplicial
trees In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are u ...
. They arise naturally in many mathematical contexts, in particular
geometric group theory Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such group (mathematics), groups and topology, topological and geometry, geometric pro ...
and
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
. They are also the simplest examples of
Gromov hyperbolic space In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations (depending quantitatively on a nonnegative real number δ) between points. The definition, introduced by Mikhael Gromov, generalizes the metric properti ...
s.


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 x, y, \rho \in X there exists a point c = x \wedge y such that the geodesic segments rho,x rho,y/math> intersect in the segment rho,c/math> and also c \in ,y/math>. 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 In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
property. A metric space X is a real tree if for any pair of points x, y \in X all
topological embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is giv ...
s \sigma of the segment ,1/math> into X such that \sigma(0) = x, \, \sigma(1) = y have the same image (which is then a geodesic segment from x to y).


Simple examples

*If X is a 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 discrete in X. * The R-tree obtained in the following way is nonsimplicial. Start with the interval , 2and glue, for each positive
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
''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 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. * More generally any
hedgehog space In mathematics, a hedgehog space is a topological space consisting of a set of spines joined at a point. For any cardinal number \kappa, the \kappa-hedgehog space is formed by taking the disjoint union of \kappa real unit intervals identified at t ...
is an example of a real tree.


Characterizations

Here are equivalent characterizations of real trees which can be used as definitions: # ''(similar to
trees In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are u ...
as graphs)'' A real tree is a
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem ...
connected space In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties tha ...
which contains no subset
homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
to a circle. # A real tree is a metric connected space (X,d) which has the four points condition (see figure) : \forall x,y,z,t\in X, d(x,y)+d(z,t)\leq \max (x,z)+d(y,t)\,;\, d(x,t)+d(y,z)/math>. # A real tree is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to connected metric space 0-hyperbolic (see figure). Formally: \forall x,y,z,t\in X, (x,y)_t\geq \min (x,z)_t\, ; \, (y,z)_t /math>. # ''(similar to the characterization of Galton-Watson trees by the contour process).'' Consider a positive excursion of a continuous function e. In other words e is such that: #* e(0)=0 #* letting \zeta(e)=\inf\be the ''end of the excursion'', we have e(t)>0 for t\in ]0,\zeta(e) /math>_ #*_e(t)=0_for_t\geq_\zeta(e)._For_x,_y\in_[0,\zeta(e)/math>,_x\leq_y,_define_a_Metric_space.html" ;"title=",\zeta(e).html" ;"title="/math> #* e(t)=0 for t\geq \zeta(e). For x, y\in [0,\zeta(e)">/math> #* e(t)=0 for t\geq \zeta(e). For x, y\in [0,\zeta(e)/math>, x\leq y, define a Metric space">pseudometric and an equivalence relation with: d_e( x, y) := e(x)+e(y)-2\min(e(z)\, ;z\in[x,y]), x\sim_e y \Leftrightarrow d_e(x,y)=0. Then, the Quotient space (topology), quotient space ([0,\zeta(e)]/\sim_e\, ,\, d_e) is a real tree. Intuitively, the
local minima In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ...
of the excursion ''e'' are the parents of the
local maxima In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ...
. 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 In probability theory, the Brownian tree, or Aldous tree, or Continuum Random Tree (CRT) is a special case from random real trees which may be defined from a Brownian excursion. The Brownian tree was defined and studied by David Aldous in three a ...
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 In mathematics, an ultralimit is a geometric construction that assigns to a sequence of metric spaces ''Xn'' a limiting metric space. The notion of an ultralimit captures the limiting behavior of finite configurations in the spaces ''Xn'' and uses ...
of a sequence (X_i) of \delta_i-
hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...
spaces with \delta_i \to 0 is a real tree. In particular, the
asymptotic cone In mathematics, an ultralimit is a geometric construction that assigns to a sequence of metric spaces ''Xn'' a limiting metric space. The notion of an ultralimit captures the limiting behavior of finite configurations in the spaces ''Xn'' and uses ...
of any hyperbolic space is a real tree.


Limit of group actions

Let G be a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
. For a sequence of based G-spaces (X_i, *_i, \rho_i) there is a notion of convergence to a based G-space (X_\infty, x_\infty, \rho_\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 = \pi_1(S) where S is a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
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 In geometric group theory, the Rips machine is a method of studying the action of groups on R-trees. It was introduced in unpublished work of Eliyahu Rips in about 1991. An R-tree is a uniquely arcwise-connected metric space in which every arc ...
. A case of particular interest is the study of degeneration of groups acting
properly discontinuously In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism g ...
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 Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
with an
ultrametric In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to d(x,z)\leq\max\left\. Sometimes the associated metric is also called a non-Archimedean metric or super-metric. Although some of the theorems ...
valuation then the
Bruhat–Tits building In mathematics, a building (also Tits building, named after Jacques Tits) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces. Bu ...
of \mathrm_2(F) 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 \Lambda = \mathbb R). There is a notion of \Lambda-tree which recovers simplicial trees when \Lambda = \mathbb Z and real trees when \Lambda = \mathbb R. The structure of
finitely presented group In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...
s 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 A building, or edifice, is an enclosed structure with a roof and walls standing more or less permanently in one place, such as a house or factory (although there's also portable buildings). Buildings come in a variety of sizes, shapes, and fun ...
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) In mathematics, a dendroid is a type of topological space, satisfying the properties that it is hereditarily unicoherent (meaning that every subcontinuum of ''X'' is unicoherent), arcwise connected, and forms a continuum. The term dendroid was in ...
* Tree-graded space


References

{{reflist Group theory Geometry Topology Trees (topology)