A note on almost negative matrices and Gromov-hyperbolic Coxeter groups

TL;DR

This paper revisits Moussong's characterization of Gromov-hyperbolic Coxeter groups, identifies a gap in the original proof, and offers a workaround to solidify the criteria based on the defining graph.

Contribution

It clarifies and corrects the proof of Moussong's characterization, enhancing understanding of hyperbolic Coxeter groups.

Findings

Identified a gap in Moussong's original proof

Provided a workaround to address the gap

Confirmed the characterization based on the defining graph

Abstract

In this note we revisit Moussong's Characterization of Gromov-hyperbolic Coxeter groups. A Coxeter group is Gromov-hyperbolic if and only if it does not contain a subgroup isomorphic to $\mathbb{Z}^2$ which can be read off directly from the defining graph. We show that there is a small gap in the original argument and provide a workaround.