Coxeter groups, hyperbolic cubes, and acute triangulations

TL;DR

This paper establishes a deep connection between abstract triangulations of the sphere, acute triangulations, and hyperbolic Coxeter groups, using CAT(-1) space theory to characterize when such triangulations correspond to hyperbolic reflection groups.

Contribution

It proves that a right-angled Coxeter group is hyperbolic if and only if the underlying triangulation is realizable as an acute triangulation, introducing a combinatorial criterion called "flag no-square."

Findings

Coxeter groups are isomorphic to hyperbolic reflection groups under acute triangulation conditions.

A triangulation of the sphere can be realized as acute if and only if it satisfies the flag no-square condition.

The results extend to other angle bounds, surfaces, and higher dimensions.

Abstract

Let $C(L)$ be the right-angled Coxeter group defined by an abstract triangulation $L$ of $\mathbb{S}^2$. We show that $C(L)$ is isomorphic to a hyperbolic right-angled reflection group if and only if $L$ can be realized as an acute triangulation. The proof relies on the theory of CAT(-1) spaces. A corollary is that an abstract triangulation of $\mathbb{S}^2$ can be realized as an acute triangulation exactly when it satisfies a combinatorial condition called "flag no-square". We also study generalizations of this result to other angle bounds, other planar surfaces and other dimensions.