Lann\'er diagrams and combinatorial properties of compact hyperbolic Coxeter polytopes

TL;DR

This paper investigates the properties of Lannér diagram products and their relation to compact hyperbolic Coxeter polytopes, establishing new classifications and bounds on polytope dimensions.

Contribution

It proves superhyperbolicity of certain Lannér diagram products and refines the maximum dimension bound for specific hyperbolic Coxeter polytopes.

Findings

All known compact hyperbolic Coxeter polytopes with product structure are classified.

Superhyperbolicity is proven for products of at least four Lannér diagrams with a diagram of order ≥ 3.

The maximum dimension of certain hyperbolic Coxeter polytopes is improved to 12.

Abstract

In this paper we study $\times_0$-products of Lann\'er diagrams. We prove that every $\times_0$-product of at least four Lann\'er diagrams with at least one diagram of order $\ge 3$ is superhyperbolic. As a corollary, we obtain that known classifications exhaust all compact hyperbolic Coxeter polytopes that are combinatorially equivalent to products of simplices. We also consider compact hyperbolic Coxeter polytopes whose every Lann\'er subdiagram has order $2$. The second result of this paper slightly improves recent Burcroff's upper bound on the dimension of such polytopes to $12$.