%0 Journal Article %T Coxeter groups, hyperbolic cubes, and acute triangulations %A Sang-hyun Kim %A Genevieve S. Walsh %J Mathematics %D 2013 %I arXiv %X 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. %U http://arxiv.org/abs/1306.6025v3