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We prove the following rank rigidity result for properCAT  ⁡   (  0  ){\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∂  ⁡  X{\partial X}    equals π and Γ does not act minimally on∂  ⁡  X{\partial X}   , then∂  ⁡  X{\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∂  ⁡  X{\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.

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