| Associated to any Coxeter system (W, S) is a labeled simplicial complex L, called the nerve of (W, S), and a complex Sigma, called the Davis complex, on which W acts properly and cocompactly. We denote the quotient space Sigma/ W by K. When L is a triangulation of Sn-1 Sigma is an n-manifold, and Sigma → Sigma/ W = K is an orbifold cover. When L is a triangulation of S1 , the geometry of Sigma is either H2 or E2 . When L = S2, K is a 3-dimensional orbifold and Sigma splits into Euclidean and non-Euclidean pieces. We call these pieces toroidal and atoroidal respectively. We examine the Euclidean cases and show that these pieces have a torus as an intermediate cover. We can also find intermediate covers for certain Coxeter system that correspond to the atoroidal pieces. However, these intermediate covers are not a classical torus. |
