Is every word-hyperbolic group CAT(0)?

Theorem 1.1: there is a finite connected two-dimensional aspherical simplicial complex $K$ with linear combinatorial disk filling, so $\pi_1K$ is word-hyperbolic with a finite classifying space, such that $\pi_1K$ has no geometric action on any proper complete CAT(0) space in any dimension. Hence it is neither CAT(0) nor CAT(-1), and no finite complex homotopy equivalent to $K$ carries a locally CAT(0) metric inducing its topology. The group comes from a random-presentation construction with a Fano-plane incidence core and a harmonic-energy fixed-point obstruction. Not shown: an explicit presentation, or which natural classes of hyperbolic groups (for example random or small-cancellation groups) are CAT(0).

Theorem 1.1: there is a finite connected two-dimensional aspherical simplicial complex $K$ with linear combinatorial disk filling, so $\pi_1K$ is word-hyperbolic with a finite classifying space, such that $\pi_1K$ has no geometric action on any proper complete CAT(0) space in any dimension. Hence it is neither CAT(0) nor CAT(-1), and no finite complex homotopy equivalent to $K$ carries a locally CAT(0) metric inducing its topology. The group comes from a random-presentation construction with a Fano-plane incidence core and a harmonic-energy fixed-point obstruction. Not shown: an explicit presentation, or which natural classes of hyperbolic groups (for example random or small-cancellation groups) are CAT(0).