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Mathematics  2013 

Coxeter groups, hyperbolic cubes, and acute triangulations

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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.


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