Geodesic
Lambert Cubes Generating Discrete Reflection Groups
Matematičeskie zametki, Tome 75 (2004) no. 2, pp. 277-286

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A Lambert cube $Q(\alpha,\beta, \gamma)$ is a combinatorial cube with dihedral angles $\alpha$, $\beta$, and $\gamma$ assigned to the three mutually noncomplanar edges and right angles at the remaining edges. In this paper, we classify the Lambert cubes in $S^3$, $\mathbb{E}^3$ 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. 
@article{MZM_2004_75_2_a10,
     author = {A. A. Felikson},
     title = {Lambert {Cubes} {Generating} {Discrete} {Reflection} {Groups}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {277--286},
     publisher = {mathdoc},
     volume = {75},
     number = {2},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2004_75_2_a10/}
}
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A. A. Felikson. Lambert Cubes Generating Discrete Reflection Groups. Matematičeskie zametki, Tome 75 (2004) no. 2, pp. 277-286. http://geodesic.mathdoc.fr/item/MZM_2004_75_2_a10/