Rank rigidity conjecture for locally compact CAT(0) spaces

Let $X$ be a locally compact geodesically complete $\operatorname{CAT}(0)$ space, and let $G$ be a discrete group acting properly and cocompactly on $X$ by isometries. Assume that $X$ is irreducible, meaning that it does not decompose as a non-trivial metric product. Rank rigidity conjecture. Then $X$ is either a higher-rank symmetric space, a Euclidean building of dimension at least $2$, or $G$ contains a rank one isometry.

The conjecture extends the rank rigidity theorem from Hadamard manifolds to broader classes of CAT(0) spaces. It is known for Euclidean cell complexes of dimensions $2$ and $3$, buildings and Coxeter groups, and finite-dimensional CAT(0) cube complexes; the general case remains open.