Lannér diagrams and combinatorial properties of compact hyperbolic Coxeter polytopes

Abstract

In this paper we study × 0 \times _0 -products of Lannér diagrams. We prove that every × 0 \times _0 -product of at least four Lannér diagrams with at least one diagram of order ≥ 3 \geq 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 Lannér subdiagram has order 2 2 . The second result of this paper slightly improves recent Burcroff’s upper bound on the dimension of such polytopes to 12 12 .