Real tree

In mathematics real trees (also called ${\displaystyle \mathbb {R} }$-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.

A metric space ${\displaystyle X}$ is a real tree if it is a geodesic space where every triangle is a tripod. That is, for every three points ${\displaystyle x,y,\rho \in X}$ there exists a point ${\displaystyle c=x\wedge y}$ such that the geodesic segments ${\displaystyle [\rho ,x],[\rho ,y]}$ intersect in the segment ${\displaystyle [\rho ,c]}$ and also ${\displaystyle c\in [x,y]}$. This definition is equivalent to ${\displaystyle X}$ 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 ${\displaystyle X}$ is a real tree if for any pair of points ${\displaystyle x,y\in X}$ all (topological) embeddings ${\displaystyle \sigma }$ of the segment ${\displaystyle [0,1]}$ into ${\displaystyle X}$ such that ${\displaystyle \sigma (0)=x,\,\sigma (1)=y}$ have the same image (which is then a geodesic segment from ${\displaystyle x}$ to ${\displaystyle y}$).

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