On equivariant homeomorphisms of boundaries of CAT(0) groups and Coxeter groups

In this paper, we investigate an equivariant homeomorphism of the boundaries ∂X and ∂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 ∂X and ∂Y   as a continuous extension of the quasi-isometry ϕ:Gx0→Gy0ϕ:Gx0→Gy0 defined by ϕ(gx0)=gy0ϕ(gx0)=gy0, where x0∈Xx0∈X and y0∈Yy0∈Y. In this paper, we say that a CAT(0) group G is 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 W1W1 and W2W2 are equivariant rigid as reflection groups, then so is W1⁎W2W1⁎W2. We also provide a conjecture on non-rigidity of boundaries of some CAT(0) groups.