Ideal right-angled polyhedra in Lobachevsky space

Abstract

In this paper we consider a class of right-angled polyhedra in three-dimensional Lobachevsky space, all vertices of which lie on the absolute. New upper bounds on volumes in terms the number of faces of the polyhedron are obtained. Volumes of polyhedra with at most 23 faces are computed. It is shown that the minimum volumes are realized on antiprisms and twisted antiprisms. The first 248 values of volumes of ideal right-angled polyhedra are presented. Moreover, the class of polyhedra with isolated triangles is introduces and there are obtained combinatorial bounds on their existence as well as minimal examples of such polyhedra are given.