Coxeter groups, hyperbolic cubes and acute triangulations

2013/06/30 by Sang-hyun Kim, S.-h. Kim, Genevieve S. Walsh +1 · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Artin group #Corollary #Coxeter group #Geometric and Algebraic Topology #Group (periodic table) #Point set triangulation #Triangulation #math.GT #msc:20F65 #msc:37F20 #msc:57M07

paper · pdf · doi:10.1112/jtopol/jtv038

published in Journal of Topology 9(1), 117-142 (Wiley) · 27 pages, 9 figures. Accepted for publication by the Journal of Topology

arxiv created 2015/06/30 · openalex publication_date 2015/12/14 · openalex created_date 2016/06/24 · arxiv updated 2019/02/07 · openalex updated_date 2026/08/05

Abstract

Let C⁡(L) be the right-angled Coxeter group defined by an abstract triangulation L of S2. 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 S2 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.

Citations

• The Colin de Verdi�re number and sphere representations of a graph
• On Nonobtuse Simplicial Partitions
• Tiling space and slabs with acute tetrahedra
• Sur un nouvel invariant des graphes et un critère de planarité
• Metric Spaces of Non-Positive Curvature
• Geometrization of 3-dimensional orbifolds
• Hyperbolic Groups