AbstractWe prove the following rank rigidity result for proper {\operatorname{CAT}(0)} spaces with one-dimensional Tits boundaries:
Let Γ be a group acting properly discontinuously, cocompactly, and by isometries on such a space X.
If the Tits diameter of {\partial X} equals π and Γ does not act minimally on {\partial X}, then {\partial X} is a spherical building or a spherical join.
If X is also geodesically complete, then X is a Euclidean building, higher rank symmetric space, or a nontrivial product.
Much of the proof, which involves finding a Tits-closed convex building-like subset of {\partial X}, does not require the Tits diameter to be π, and we give an alternate condition that guarantees rigidity when this hypothesis is removed, which is that a certain invariant of the group action be even.