Abstract
Let X be a proper CAT(0) space and G a group of isometries of X acting cocompactly without fixed point at infinity. We prove that if \partial X contains an invariant subset of circumradius \pi/2 , then X contains a quasi-dense, closed convex subspace that splits as a product. Adding the assumption that the G -action on X is properly discontinuous, we give more conditions that are equivalent to a product splitting. In particular, this occurs if \partial X contains a proper nonempty, closed, invariant, \pi -convex set in \partial X ; or if some nonempty closed, invariant set in \partial X intersects every round sphere K \subset \partial X inside a proper subsphere of K .
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