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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

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    $\begingroup$ It is an open problem if every hyperbolic group acts isometrically properly dis continuously and cocompactly on a CAT(-1) space. But every hyperbolic group is quasi-isometric to such space. $\endgroup$
    – Misha
    Commented Jun 27, 2019 at 22:09

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

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 $P_d(G)$?

This question is also open, but a counterexample is given here (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.

Actually, even the following question is open:

Let $G$ be a hyperbolic group acting geometrically on a CAT(0) cube complex. Does $G$ act geometrically on a CAT(-1) cube complex?

The question appears for instance here.

Also relevant is this article of Brady and Crips constructing a hyperbolic group with CAT(0) dimension two but CAT(-1) dimension three.

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  • $\begingroup$ Thank you very much, this is very interesting. $\endgroup$
    – spiramirabilis
    Commented Jun 28, 2019 at 14:21

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