# Real tree and hyperbolicity

I seek a proof of the following result due to Tits:

Theorem: A path-connected $0$-hyperbolic metric space is a real tree.

Do you know any proof or reference?

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What is the definition of "real tree" you are using? This is usually taken as the definition! –  user641 Jan 18 '13 at 17:04
@SteveD: According to wikipedia, a real tree is a metric space $(M,d)$ such that for any $x, y$ in $M$ there is a unique arc from $x$ to $y$ and this arc is a geodesic segment. I use the same definition. –  Seirios Jan 18 '13 at 21:44

I finally found the theorem in a document by Steven N. Evans: Probability and Real Trees (theorem 3.40).

Moreover, path-connected can be replaced by connected.

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Thanks for sharing! –  Dimitrios Nt Oct 23 '14 at 17:50

Might this be what you are looking for?

J. Tits, A "theorem of Lie-Kolchin" for trees, Contributions to Algebra: A Collection of Papers Dedicated to Ellis Kolchin, Academic Press, New York, 1977, pp. 377–388.

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Unfortunately, the article does not contain the above result. –  Seirios Jan 23 '13 at 18:16
@Seirios Please accept my apologies for having sent you in the wrong direction. –  Glen The Udderboat Jan 23 '13 at 20:37
Even so, I found other interesting things! –  Seirios Jan 23 '13 at 21:03