The number of cusps of right-angled polyhedra in hyperbolic spaces

As was pointed out by Nikulin [8] and Vinberg [10], a right-angled polyhedron of finite volume in the hyperbolic n-space Hⁿ has at least one cusp for n ≥ 5. We obtain non-trivial lower bounds on the number of cusps of such polyhedra. For example, right-angled polyhedra of finite volume must have at least three cusps for n = 6. Our theorem also says that the higher the dimension of a right-angled polyhedron becomes, the more cusps it must have.