Euclidean simplices generating discrete reflection groups

Let P be a convex polytope in the spherical space S, in the Euclidean space E, or in the hyperbolic space H. Consider the group GP generated by reflections in the facets of P . We call GP a reflection group generated by P . The problem we consider in this paper is to list polytopes generating discrete reflection groups. The answer is known only for some combinatorial types of polytopes. Already in 1873, Schwarz [10] listed spherical triangles generating discrete groups. In 1998, E. Klimenko and M. Sakuma [9] solved the problem for hyperbolic triangles. In [2], [4], [3], [5] the problem was solved for hyperbolic quadrilaterals, compact hyperbolic pyramids and triangular prisms, hyperbolic simplices, and Lambert cubes in S, E, H. In [6] the problem was solved for spherical simplices.