# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1902.01029