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

Abstract

Abstract We prove the following rank rigidity result for proper CAT ⁡ ( 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.