On hyperbolic Coxeter n-cubes

Beside simplices, n-cubes form an important class of simple polyhedra. Unlike hyperbolic Coxeter simplices, hyperbolic Coxeter n-cubes are not classified. In this work, we first show that there are no Coxeter n-cubes in H for n ≥ 10. Then, we show that the ideal ones exist only for n = 2 and 3, 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.