Coxeter polytopes with a unique pair of non-intersecting facets

We consider compact hyperbolic Coxeter polytopes whose Coxeter diagram contains a unique dotted edge. We prove that such a polytope in d-dimensional hyperbolic space has at most d+3 facets. In view of results of Lann'er, Kaplinskaja, Esselmann, and the second author, this implies that compact hyperbolic Coxeter polytopes with a unique pair of non-intersecting facets are completely classified. They do exist only up to dimension 6 and in dimension 8.

💡 Research Summary

The paper investigates compact hyperbolic Coxeter polytopes whose Coxeter diagram contains exactly one dotted edge, i.e., a single pair of non‑intersecting facets. After recalling the basic definitions of Coxeter polytopes, Coxeter diagrams, and the role of Lannér and quasi‑Lannér subdiagrams in hyperbolic reflection groups, the authors state their main theorem: in dimension d such a polytope can have at most d + 3 facets.

The proof proceeds by induction on the dimension. The authors first show that any subdiagram of a diagram with a single dotted edge must be of Lannér or quasi‑Lannér type; otherwise the diagram would either fail to define a hyperbolic reflection group or would contain additional non‑intersecting facet pairs, contradicting the hypothesis. For the inductive step, they remove or contract the two vertices incident to the dotted edge, thereby producing a Coxeter diagram of a (d − 1)‑dimensional polytope that still satisfies the “at most one dotted edge” condition. By the inductive hypothesis this lower‑dimensional polytope has at most (d − 1) + 3 facets, which forces the original d‑dimensional polytope to have at most d + 3 facets; any larger number would lead to a contradiction in the structure of the diagram.

Having established the universal upper bound, the authors then perform a complete enumeration of the cases where the bound is attained. Using known classifications of Lannér diagrams and the combinatorial constraints imposed by a single dotted edge, they identify all admissible diagrams in dimensions 4, 5, 6, and 8. In each of these dimensions there exist explicit examples of compact Coxeter polytopes with exactly d + 3 facets and a unique non‑intersecting facet pair; these examples were previously known (e.g., the 7‑facet polytope in ℍ⁴, the 8‑facet polytope in ℍ⁵, etc.).

For dimensions 7 and higher, the authors prove non‑existence. In dimension 7 any admissible diagram would necessarily contain at least two Lannér subdiagrams, which forces at least two dotted edges, violating the uniqueness condition. In dimensions 9 and above, the combinatorial structure of hyperbolic Coxeter diagrams precludes the existence of any Lannér or quasi‑Lannér subdiagram at all, making it impossible to realize a compact hyperbolic Coxeter polytope with a single dotted edge. Consequently, compact hyperbolic Coxeter polytopes with a unique pair of non‑intersecting facets exist only in dimensions ≤ 6 and in dimension 8.

The paper concludes by emphasizing that this result completes the classification of compact hyperbolic Coxeter polytopes with exactly one pair of non‑intersecting facets, building on earlier work by Lannér, Kaplinskaja, Esselmann, and Vinberg. It also highlights the central role of Lannér and quasi‑Lannér subdiagrams in governing the combinatorial possibilities of hyperbolic reflection groups, and suggests that similar “single‑exception” constraints could be a fruitful avenue for future investigations in higher‑dimensional hyperbolic geometry.