Geometrization of 3-dimensional Coxeter orbifolds and Singer’s conjecture

Abstract

Associated to any Coxeter system (W, S), there is a labeled simplicial complex L and a contractible CW-complex Σ L (the Davis complex) on which W acts properly and cocompactly. Σ L admits a cellulation under which the nerve of each vertex is L. It follows that if L is a triangulation of \({\mathbb{S}^{n-1}}\), then Σ L is a contractible n-manifold. In this case, the orbit space, K L := Σ L /W, is a Coxeter orbifold. We prove a result analogous to the JSJ-decomposition for 3-dimensional manifolds: Every 3-dimensional Coxeter orbifold splits along Euclidean suborbifolds into the characteristic suborbifold and simple (hyperbolic) pieces. It follows that every 3-dimensional Coxeter orbifold has a decomposition into pieces which have hyperbolic, Euclidean, or the geometry of \({\mathbb{H}^2\,\times\,\mathbb{R}}\). (We leave out the case of spherical Coxeter orbifolds.) A version of Singer’s conjecture in dimension 3 follows: That the reduced ℓ 2-homology of Σ L vanishes.