CAT(0) spaces of higher rank II

This belongs to a series of papers motivated by Ballmann’s Higher Rank Rigidity Conjecture. We prove the following. Let \(X\) be a CAT(0) space with a geometric group action \(\Gamma \curvearrowright X\). Suppose that every geodesic in \(X\) lies in an \(n\)-flat, \(n\geq 2\). If \(X\) contains a periodic \(n\)-flat which does not bound a flat \((n+1)\)-half-space, then \(X\) is a Riemannian symmetric space, a Euclidean building or non-trivially splits as a metric product. This generalizes the Higher Rank Rigidity Theorem for Hadamard manifolds with geometric group actions.