A rank rigidity result for CAT(0) spaces with one-dimensional Tits boundaries

We prove the following rank rigidity result for proper CAT(0) spaces with one-dimensional Tits boundaries: Let $\mathrm{\Gamma }$ be a group acting properly discontinuously, cocompactly, and by isometries on such a space $X$. If the Tits diameter of $\mathrm{\partial }X$ equals $\pi$ and $\mathrm{\Gamma }$ does not act minimally on $\mathrm{\partial }X$, then $\mathrm{\partial }X$ 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 $\mathrm{\partial }X$, does not require the Tits diameter to be $\pi$, 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.