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 Gamma be a group acting properly discontinuously, cocompactly, and by isometries on such a space X. If the Tits diameter of partial derivative X equals pi and Gamma does not act minimally on partial derivative X, then partial derivative 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 partial derivative 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.