Geometry of right-angled Coxeter groups on the Croke-Kleiner spaces

TL;DR

This paper explores the geometric actions of right-angled Coxeter groups on Croke-Kleiner spaces, revealing rigidity in their angle configurations compared to quasi-isometric right-angled Artin groups.

Contribution

It demonstrates that right-angled Coxeter groups acting on these spaces have fixed $rac{ ext{pi}}{2}$ angles, highlighting their geometric rigidity versus the flexibility of related Artin groups.

Findings

Coxeter groups act with $rac{ ext{pi}}{2}$ angles on Croke-Kleiner spaces.

Quasi-isometric Artin groups can act with any angle in $(0, rac{ ext{pi}}{2}]$.

Coxeter groups exhibit greater geometric rigidity than their Artin counterparts.

Abstract

In this paper we study the right-angled Coxeter groups that acts geometrically on the Salvetti complex of a certain right-angled Artin group, which we refer to as Croke-Kleiner spaces. We prove that any right-angled Coxeter group that acts geometrically on the Croke-Kleiner spaces acts with $\pi/2$ angles between reflecting axes, while the quasi-isometric right-angled Artin group can act with angles that are any real number in the range $(0, \pi/2]$. The contrast between the two examples shows that in this case a right-angled Coxeter group is geometrically more "rigid" than its quasi isometric counterpart.