On splitting theorems for CAT(0) spaces and compact geodesic spaces of non-positive curvature

In this paper, we show some splitting theorems for CAT(0) spaces on which a product group acts geometrically and we obtain a splitting theorem for compact geodesic spaces of non-positive curvature. A CAT(0) group Γ is said to be rigid, if Γ determines its boundary up to homeomorphisms of a CAT(0) space on which Γ acts geometrically. C. Croke and B. Kleiner have constructed a non-rigid CAT(0) group. As an application of the splitting theorems for CAT(0) spaces, we obtain that if Γ₁ and Γ₂ are rigid CAT(0) groups then so is Γ₁ × Γ₂.