It is mentioned on the Wikipedia article for Hadamard spaces that the Cayley graphs of a word-hyperbolic (f.g.) group are CAT(0) metric spaces. Is it so? My question comes from the fact that the Cayley graph for S$L(2,\mathbb{Z})$ with the two usual generators $z\mapsto z+1$ and $z\mapsto -z^{-1}$ is very tree-like but still has 1-sided equilateral triangles (whose inner distances are obviously larger than those on Euclidean space).

At the same time, if it were true, then any word-hyperbolic group would act properly and cocompactly on its Cayley graph, and we would have an affirmative answer to this very similar (and as far as it seems, open) question. Am I missing something? (I tend to assume that general math articles on the English Wikipedia are correct, yes)

  • $\begingroup$ But a Hadamard space is a complete CAT(0) space (at least that's the definition I have from Ballmann's lectures, Wikipedia has the midpoint inequality definition and then says they are equivalent)... $\endgroup$ Apr 8, 2016 at 19:19
  • $\begingroup$ It looks like a mistake, I looked at the history and it used to say cayley graph of discrete groups, which is certainly not true, and I guess the person who corrected that ended up inserting a more subtle error, although it is a well known open problem if hyperbolic groups are CAT(0) (note there is no reference to the claim). $\endgroup$
    – user29123
    Apr 8, 2016 at 20:46
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    $\begingroup$ A graph which is not a tree, equipped with a geodesic metric putting length 1 on each edge as Cayley graphs are, is certainly is not CAT(0), for example any minimal length embedded circle in the graph can be used to disprove the CAT(0) inequality. So the only groups that can have a CAT(0) Cayley graph are free groups. $\endgroup$
    – Lee Mosher
    Apr 9, 2016 at 0:42
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    $\begingroup$ Alternatively, CAT(0) spaces are simply connected (in fact, contractible), so trees are the only CAT(0) graphs. $\endgroup$
    – Seirios
    Apr 9, 2016 at 9:59
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    $\begingroup$ I edited the Wikipedia page, and edited the link on this question to go to the old version to preserve the question (just letting future people who see this question know). $\endgroup$
    – user29123
    Apr 9, 2016 at 16:57

1 Answer 1


This question has been answered by Lee Mosher and Seirios in the comments. The only graphs that are CAT(0) are trees, so the groups whose Cayley graphs are CAT(0) are free groups.


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