# Nonhyperbolic Coxeter groups with Menger curve boundary (with erratum)

## TL;DR

This paper constructs the first known examples of non-hyperbolic CAT(0) groups with a Menger curve boundary, expanding understanding of boundaries in geometric group theory.

## Contribution

It introduces non-hyperbolic CAT(0) Coxeter groups with Menger curve boundaries, extending the class of groups known to have such boundaries.

## Key findings

- First examples of non-hyperbolic CAT(0) groups with Menger curve boundary
- Construction based on Coxeter groups with complete graph nerves
- Extension of Sierpiński's theorem used in the construction

## Abstract

A generic finite presentation defines a word hyperbolic group whose boundary is homeomorphic to the Menger curve. In this article, we produce the first known examples of non-hyperbolic $CAT(0)$ groups whose visual boundary is homeomorphic to the Menger curve. The examples in question are the Coxeter groups whose nerve is a complete graph on $n$ vertices for $n\geq 5$. The construction depends on a slight extension of Sierpi\'nski's theorem on embedding $1$--dimensional planar compacta into the Sierpi\'nski carpet. See the appendix for a brief erratum.