On hyperbolic Coxeter n-cubes

Abstract

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 n 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.

All hyperbolic Coxeter n-cubes

“…• The classification described in Table 2 contains several already available partial classifications (see [1,7,8], for example).…”

“…However, this exclusion strategy (which has been successfully applied in the particular case treated in [8]), in contrast to the Gröbner basis extraction algorithm, cannot be made systematic, as there is no satisfactory method to decide how the sets S of indices have to be chosen, and how many of such sets are sufficient to yield an impossible setting.…”

“…In a previous paper [8], the first author showed that there are no hyperbolic Coxeter n-cubes in H n for n ≥ 10, and a combinatorial/algebraic approach has been used in order to classify the ideal hyperbolic Coxeter n-cubes, which exist only for n ≤ 3. As a consequence of the relatively low dimensional upper bound and the explicit approach leading to the classification in the ideal case, it was finally suggested that a computer assisted approach could successfully handle the large amount of combinatorial possibilities in the general setting, and therefore proceed with the classification.…”

“…In this paper, we follow this strategy and eventually provide a full classification of all hyperbolic Coxeter n-cubes. To this end, we refine and expand the combinatorial and algebraic methods from [8], and we implement them with help of Mathematica.…”

On the commensurability of hyperbolic Coxeter groups

“…Example 3.19. As an example, consider the two non-cocompact quasi-arithmetic (but not arithmetic) Coxeter cube groups Γ 1 and Γ 2 in Isom(H 3 ) defined in Figure 3 (see [18]). They both have Q as Vinberg field and similar quadratic forms.…”