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

TL;DR

This paper explores conditions under which boundaries of CAT(0) spaces with group actions are equivariantly homeomorphic, introduces the concept of equivariant boundary rigidity, and examines rigidity properties of Coxeter groups and their free products.

Contribution

It establishes criteria for equivariant boundary homeomorphisms, defines equivariant boundary rigidity for CAT(0) groups, and analyzes rigidity properties of Coxeter groups and their free products.

Findings

Provided sufficient and equivalent conditions for boundary homeomorphisms

Introduced the concept of equivariant boundary rigidity

Showed that rigidity is preserved under free products of Coxeter groups

Abstract

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.