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 $\Sigma_L$ (the Davis complex) on which $W$ acts properly and cocompactly. $\Sigma_L$ 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 $\Sigma_L$ is a contractible $n$-manifold. In this case, the orbit space, $K_L:=\Sigma_L/W$, is a \emph{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 \emph{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 $\ell^2$-homology of $\Sigma_L$ vanishes.