On simple ideal hyperbolic Coxeter polytopes

Let IH be the n-dimensional hyperbolic space and let P be a simple polytope in IH. P is called an ideal polytope if all vertices of P belong to the boundary of IH. P is called a Coxeter polytope if all dihedral angles of P are submultiples of π. There is no complete classification of hyperbolic Coxeter polytopes. In [6] Vinberg proved that there are no compact hyperbolic Coxeter polytopes in IH when n ≥ 30. Prokhorov [5] and Khovanskij [3] proved that there are no Coxeter polytopes of finite volume in IH for n ≥ 996. Examples of bounded Coxeter polytopes are known only for n ≤ 8, and examples of finite volume non-compact Coxeter polytopes are known only for n ≤ 19 [8] and n = 21 [1]. In this paper, we prove that no simple ideal Coxeter polytope exists in IH when n > 8. The authors are grateful to the Max-Planck Institute for Mathematics in Bonn for hospitality and excellent research conditions.