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 $\Gamma$ is said to be {\it rigid}, if $\Gamma$ determines the boundary up to homeomorphisms of a CAT(0) space on which $\Gamma$ 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 $\Gamma_1$ and $\Gamma_2$ are rigid CAT(0) groups then so is $\Gamma_1\times \Gamma_2$.Comment: 14 page

On equivariant homeomorphisms of boundaries of CAT(0) groups

In this paper, we investigate an equivariant homeomorphism of the boundaries $\partial X$ and $\partial Y$ of two proper CAT(0) spaces $X$ and $Y$ on which a CAT(0) group $G$ acts geometrically. We provide a sufficient condition to obtain a $G$-equivariant homeomorphism of the two boundaries $\partial X$ and $\partial Y$ as a continuous extension of the quasi-isometry $\phi:Gx_0\to Gy_0$ defined by $\phi(gx_0)=gy_0$, where $x_0\in X$ and $y_0\in Y$.Comment: 15 page