On hyperbolic Coxeter nn-cubes

Besides simplices, nn-cubes form an important class of simple polyhedra. Unlike hyperbolic Coxeter simplices, hyperbolic Coxeter nn-cubes are not classified. In this work, we first show that there are no Coxeter nn-cubes in HnHn for n≥10n≥10. Then, we show that the ideal   ones exist only for n=2n=2 and 33, and provide a classification. The methods used are of combinatorial and algebraic nature, using properties of a Coxeter graph, its Schläfli matrix, and the Gram matrix of a polyhedron.