Mikhail Gromov states that he "tried for about 10 years to prove that every hyperbolic group is realizable by a space of negative curvature" in his interview with Martin Raussen and Christian Skau (page 6, link below). Is there any information available about the current status of this conjecture?


<https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/rtx100300391p.pdf>



Hyperbolic groups and spaces of negative curvature
As already mentioned in the comments, it is still unknown whether hyperbolic groups are CAT(-1) or even CAT(0). A related question is:



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> Let $G$ be a hyperbolic group (endowed with a finite generated set). If $d$ is sufficiently large, does there exist a $G$-equivariant CAT(-1) or CAT(0) metric defined on the Rips complex $P\_d(G)$?
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This question is also open, but a counterexample is given [here](https://arxiv.org/abs/1006.1939) (Corollary 5.10) for $P\_d(X)$ where $X$ is a specific (locally infinite) quasi-tree.


An alternative approach is to replace Rips' complex with the injective hull of the group. See the last paragraph of the introduction of Lang's [paper](https://arxiv.org/abs/1107.5971).


Actually, even the following question is open:



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> Let $G$ be a hyperbolic group acting geometrically on a CAT(0) cube complex. Does $G$ act geometrically on a CAT(-1) cube complex?
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The question appears for instance [here](https://sites.google.com/a/scu.edu/rscott/pggt).


Also relevant is this [article](https://www2.math.ou.edu/~nbrady/papers/cathyp2.pdf) of Brady and Crips constructing a hyperbolic group with CAT(0) dimension two but CAT(-1) dimension three.


