On equivariant homeomorphisms of boundaries of CAT(0) groups

TL;DR

This paper establishes conditions under which boundaries of two proper CAT(0) spaces, acted upon by a group, are homeomorphic in a way that respects the group action, extending a quasi-isometry.

Contribution

It provides a sufficient condition for a G-equivariant homeomorphism between the boundaries of two CAT(0) spaces, extending a given quasi-isometry.

Findings

Identifies conditions for boundary homeomorphisms to be G-equivariant

Extends quasi-isometries to boundary homeomorphisms

Enhances understanding of boundary structures in CAT(0) geometry

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 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 $y_0\in Y$.