The Number of Cusps of Right-angled Polyhedra in Hyperbolic Spaces

Abstract

As was pointed out by Nikulin [8] and Vinberg [10], a right-angled polyhedron of finite volume in the hyperbolic $n$-space $\mathbf{H}^n$ has at least one cusp for $n\geq 5$. 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.