. M G ] 1 1 Ju n 20 04 Hyperbolic Coxeter n - polytopes with n + 3 facets

A polytope is called a Coxeter polytope if its dihedral angles are integer parts of π. In this paper we prove that if a noncompact Coxeter polytope of finite volume in IH has exactly n+3 facets then n ≤ 16. We also find an example in IH and show that it is unique. 1. Consider a convex polytope P in n-dimensional hyperbolic space IH. A polytope is called a Coxeter polytope if its dihedral angles are integer parts of π. Any Coxeter polytope P is a fundamental domain of the discrete group generated by the reflections with respect to facets of P . Of special interest are hyperbolic Coxeter polytopes of finite volume. In contrast to spherical and Euclidean cases there is no complete classification of such polytopes. It is known that dimension of compact Coxeter polytope does not exceed 29 (see [12]), and dimension of non-compact polytope of finite volume does not exceed 995 (see [9]). Coxeter polytopes in IH are completely characterized by Andreev [1], [2]. There exists a complete classification of hyperbolic simplices [8], [11] and hyperbolic n-polytopes with n+ 2 facets [7] (see also [13]), [5], [10]). In [5] Esselmann proved that n-polytopes with n + 3 facets do not exist in IH, n > 8, and the example found by Bugaenko [3] in IH is unique. There is an example of finite volume non-compact polytope in IH with 18 facets (see [13]). The main result of this note is the following theorem: Theorem 1. Dimension of finite volume non-compact hyperbolic Coxeter n-polytope with n + 3 facets does not exceed 16. In IH there exists a unique polytope with 19 facets; its Coxeter diagram is presented below. 2. To represent Coxeter polytopes one can use Coxeter diagrams. Nodes of Coxeter diagram correspond to facets of polytope. Two nodes are joined by a (m−2)-fold edge or am-labeled edge if the corresponding dihedral angle equals π m . If the corresponding facets are parallel the nodes are joined by a bold edge, and if they diverge then the nodes are joined by a dotted edge labeled by cosh(ρ), where ρ is the distance between the facets. Every combinatorial type of n-polytope with n + 3 facets can be represented by a standard two-dimensional Gale diagram (see, for example, [6]). This consists of vertices of regular 2k-gon in IR centered at the origin and (possibly) the origin which are labeled according to the following rules: 1) Each label is a non-negative integer, the sum of labels equals n+ 3. 2) Labels of neighboring vertices can not be equal to zero simultaneously.