Lambert Cubes Generating Discrete Reflection Groups

A Lambert cube Q(α,β,γ) is a combinatorial cube with dihedral angles α, β, and γ assigned to the three mutually noncomplanar edges and right angles at the remaining edges. In this paper, we classify the Lambert cubes in S³, $\mathbb{E}^3 $ , and $\mathbb{H}^3 $ such that the group G ^Q generated by the reflections with respect to the faces of a cube Q is discrete.