The effect of dimension 4 in Aleksandrov's problem of filling a space by polyhedra

This paper deals with filling the hyperbolic space Hⁿ by non-compact polyhedra. In dimensions n <4 the non-compact case is very different from the compact one, which was investigated by A.D. Aleksandrov. For n 4 the compact and non-compact cases are almost similar. This investigation is closely related to deformations of complete and incomplete hyperbolic orbifolds (in the sense of W. Thurston) for which a strong rigidity result is proved-similar to the one for complete hyperbolic manifolds in dimension exceeding two.