On equivariant homeomorphisms of boundaries of CAT(0) groups and Coxeter 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 and an equivalent condition to obtain a $G$-equivariant homeomorphism of the boundaries $\partial X$ and $\partial Y$ as a continuous extension of the quasi-isometry $\phi:Gx_0\rightarrow Gy_0$ defined by $\phi(gx_0)=gy_0$, where $x_0\in X$ and $y_0\in Y$. In this paper, we say that a CAT(0) group $G$ is {\it equivariant (boundary) rigid}, if $G$ determines its ideal boundary by the equivariant homeomorphisms as above. As an application, we introduce some examples of (non-)equivariant rigid CAT(0) groups and we show that if Coxeter groups $W_1$ and $W_2$ are equivariant rigid as reflection groups, then so is $W_1 * W_2$. We also provide a conjecture on non-rigidity of boundaries of some CAT(0) groups.