In mathematics, real trees (also called R {\displaystyle \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 {\displaystyle 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 , ρ ∈ X {\displaystyle x,y,\rho \in X} there exists a point c = x ∧ y {\displaystyle c=x\wedge y} such that the geodesic segments [ ρ , x ] , [ ρ , y ] {\displaystyle [\rho ,x],[\rho ,y]} intersect in the segment [ ρ , c ] {\displaystyle [\rho ,c]} and also c ∈ [ x , y ] {\displaystyle c\in [x,y]} . This definition is equivalent to X {\displaystyle 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 {\displaystyle X} is a real tree if for any pair of points x , y ∈ X {\displaystyle x,y\in X} all topological embeddings σ {\displaystyle \sigma } of the segment [ 0 , 1 ] {\displaystyle [0,1]} into X {\displaystyle X} such that σ ( 0 ) = x , σ ( 1 ) = y {\displaystyle \sigma (0)=x,\,\sigma (1)=y} have the same image (which is then a geodesic segment from x {\displaystyle x} to y {\displaystyle y} ).
Simple examples If X {\displaystyle 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 {\displaystyle T} is simplicial if and only if the set of singular points of X {\displaystyle X} (points whose complement in X {\displaystyle X} has three or more connected components) is closed and discrete in X {\displaystyle X} . The R {\displaystyle \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 R {\displaystyle \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 {\displaystyle P} , and if two points are on the same ray from P {\displaystyle 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 {\displaystyle 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.
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 ) {\displaystyle (X,d)} which has the four points condition (see figure):
For all x , y , z , t ∈ X , {\displaystyle x,y,z,t\in X,} d ( x , y ) + d ( z , t ) ≤ max [ d ( x , z ) + d ( y , t ) ; d ( x , t ) + d ( y , z ) ] {\displaystyle d(x,y)+d(z,t)\leq \max[d(x,z)+d(y,t)\,;\,d(x,t)+d(y,z)]} . 3) A real tree is a connected 0-hyperbolic metric space (see figure). Formally,
For all x , y , z , t ∈ X , {\displaystyle x,y,z,t\in X,} ( x , y ) t ≥ min [ ( x , z ) t ; ( y , z ) t ] , {\displaystyle (x,y)_{t}\geq \min[(x,z)_{t}\,;\,(y,z)_{t}],}
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