Right-angled Coxeter groups, polyhedral complexes, and acute triangulations (dans le cadre de l'atelier Autour de la conjecture de Cannon )

Given a (combinatorial) triangulation T of the two-sphere, there is a right-angled coxeter group C(T) which is defined by the one-skeleton of T. When the triangulation T can be realized as an acute triangulation, we show how to build a CAT(-1) polyhedral complex on which C(T) acts geometrically. This space is quasi-isometric to $\mathbb{H}^3$. As a consequence, a triangulation of the two-sphere can be realized as an acute triangulation if and only if it does not contain any separating 3- or 4- cycles. This is joint work with Sang-hyun Kim, KAIST.