Right-angled Coxeter groups with non-planar boundary

TL;DR

This paper explores the relationship between the defining graphs of right-angled Coxeter groups and the planarity of their boundaries, revealing that non-planar graphs do not always lead to non-planar group boundaries.

Contribution

It provides a nuanced analysis showing that non-planarity of the defining graph does not always determine the boundary's planarity, and characterizes graphs with Menger curve boundaries.

Findings

Non-planarity of the defining graph does not always imply non-planarity of the boundary.

Certain triangle-free graphs produce boundaries homeomorphic to the Menger curve.

The paper offers a characterization of graphs whose Coxeter group boundaries are Menger curves.

Abstract

We investigate the planarity of the boundaries of right-angled Coxeter groups. We show that non-planarity of the defining graph does not necessarily imply non-planarity of every boundary of the associated right-angled Coxeter group, although it does in many cases. Our techniques yield a characterization of the triangle-free defining graphs such that the associated right-angled Coxeter group has boundary a Menger curve.