Coxeter groups, hyperbolic cubes, and acute triangulations

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.

27 pages, 9 figures. Accepted for publication by the Journal of Topology

References in corpus (2)

• Tiling space and slabs with acute tetrahedra
• Well-Centered Triangulation