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Let $X$ be a geodesic space. Then the following conditions are equivalent:

  • For any $x,y\in X, x \neq y$, there is a unique arc (homeomorphic to the interval $[0,1]$) with endpoints $x$ and $y$.
  • No subset of $X$ is homeomorphic to $S^1$.
  • $X$ is simply connected and its topological dimension (small inductive) equals $1$.
  • Every geodesic triangle is isometric to a tripod.
  • $X$ is $0$-hyperbolic in the sense of Gromov.
  • Intersection of any two closed balls is a closed ball or an empty set.
  • For every Lipschitz maps $\gamma:S^1\to X$ and $\pi:X\to\mathbb{R}^2$, $$ \int_{S^1}(\pi\circ\gamma)^*(x\, dy)=0. $$

A metric space that satisfies any of the above equivalent conditions is known as the metric tree or an $\mathbb{R}$-tree.

The problem is that it is very difficult to find a single place where one could find proofs of such equivalences. Thus my question is:

Question. Is there a single paper that where one would find seld-contained proofs of such characterizations of metric trees? I do not necessarily mean exactly the same characterizations. Just many equivalent characterizations.

I wanted to ask my student to write a survey paper that would cover in particular all such proofs. I think it would be a useful reference for people working in analysis on metric spaces, and for him it would be a good way to learn this material, but if there is already a good reference for such characterizations, then perhaps it would be a pointless task. This is why I am asking that question.

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    $\begingroup$ If you mean PhD student, I don't think it is the right person to ask to write such a survey. In any case, you might look at references in ams.org/journals/tran/1992-334-01/S0002-9947-1992-1081940-X/… (Mayer, Nikiel, Oversteegen, T.AMS 1992) $\endgroup$
    – YCor
    May 22, 2021 at 14:36
  • $\begingroup$ @YCor Thank you for the reference. Yes, a PhD student, but I would help him so it would likely be a joint work. I know it is not a task for someone unexperienced, but a good way to learn a lot of topics. $\endgroup$
    – Piotr Hajlasz
    May 22, 2021 at 15:05
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    $\begingroup$ @YCor I agree, but he just started and this is a good way to learn how to write and extract information from the papers. Depending how it goes it might be just a part of his thesis and not a paper. It is not intended to be his research. Just a way to learn. Anyway, I truly hate that people tend to publish useless technical "new" results and rarely try to write useful "service" papers that ar focused on a readable presentation of what is known. $\endgroup$
    – Piotr Hajlasz
    May 22, 2021 at 15:50
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    $\begingroup$ "Anyway, I truly hate that people tend to publish useless technical "new" results and rarely try to write useful "service" papers"--- hear, hear. $\endgroup$
    – Nik Weaver
    May 22, 2021 at 17:48
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    $\begingroup$ As mentionned a survey of these results isn't really paper-worthy, but I still think it's an excellent exercice for a grad student to work through. I also think this could be a good entry in a math blog or other less formal publication. $\endgroup$
    – NWMT
    May 23, 2021 at 13:26

1 Answer 1

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Some equivalent characterizations of metric trees can be extracted from Theorem 8 of the paper "Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: A note on combinatorial properties of metric spaces" by Dress (1984) DOI link. The focus of the theorem is more on subsets of metric trees and injective hulls/tight spans, but the proof is self-contained.

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  • $\begingroup$ Since you are an expert in geometric measure theory and topology, do you know if the result mathoverflow.net/q/391839/121665 is known? The statement is nice, I think, and I would like to know if it is something new. $\endgroup$
    – Piotr Hajlasz
    May 24, 2021 at 20:25
  • $\begingroup$ Since you are an expert in metric trees and at some point you had to learn it from scratch, do you think such a survey paper (assuming it is really well written, not just copy and paste) would make sense? I suspect that when writing it, some of the proofs would be modified and presented in a more readable form than the original ones. Just be honest. $\endgroup$
    – Piotr Hajlasz
    May 28, 2021 at 12:29
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    $\begingroup$ I don't know if the result you linked above is known. I actually would have benefited from such a survey paper at some point, so I think it is a good idea. I couldn't find a single paper that does the trick and collected the results I needed from different sources. Sure, many equivalences are implied by more general results about spaces with upper curvature bounds and contained in the book of Bridson and Haefliger for example, but I think trees are interesting enough to have them singled out with self-contained concise proofs. $\endgroup$
    – rozu
    May 29, 2021 at 12:13

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