Geometric CAT(0) actions for hyperbolic groups

Determine whether every Gromov-hyperbolic group acts geometrically on a CAT(0) space.

Background

The notes contrast injective spaces with CAT(0) spaces and recall that Gromov-hyperbolic groups are known to act geometrically on injective metric spaces. The corresponding CAT(0) action problem is presented as a longstanding unresolved problem in geometric group theory.

References

Whereas it is a notorious open problem whether Gromov-hyperbolic groups act geometrically on CAT(0) spaces, Lang proves that Gromov-hyperbolic groups act geometrically on injective metric spaces.

— Group actions on injective spaces and Helly graphs  (2307.00414 - Haettel, 2023) in Section 1, Introduction