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

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 $12$.