A splitting theorem for $CAT(0)$ spaces with the geodesic extension property

Abstract

In this paper, we show the following splitting theorem: For a proper $\mathrm{CAT}(0)$ space $X$ with the geodesic extension property, if a group $\Gamma=G_{1}\times G_{2}$ acts geometrically (i.e., properly discontinuously and cocompactly by isometries) on $X$, then $X$ splits as a product $X_{1}\times X_{2}$ and there exist geometric actions of $G_1$ and some subgroup of finite index in $G_2$ on $X_1$ and $X_2$, respectively.