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)
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.