Sang-hyun Kim, Acute Triangulations of the Sphere

We prove that a combinatorial triangulation L of a sphere admits an acute geodesic triangulation if and only if L does not have a separating three- or four-cycle. The backward direction is an easy consequence of the Andreev–Thurston theorem on orthogonal circle packings. For the forward direction, we consider the Davis manifold M from L. The acuteness of L will provide M with a CAT(-1) (hence, hyperbolic) metric. As a non-trivial example, we show the non-existence of an acute realization for an abstract triangulation suggested by Oum; the degrees of the vertices in that triangulation are all larger than four. This approach generalizes to triangulations coming from more general Coxeter groups, and also to planar triangulations. (Joint work with Genevieve Walsh)