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

Abstract

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 {\it rigid}, if $Γ$ determines the 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 $Γ_1$ and $Γ_2$ are rigid CAT(0) groups then so is $Γ_1\times Γ_2$.