On simple ideal hyperbolic Coxeter polytopes

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...

Essential Hyperbolic Coxeter Polytopes

We introduce a notion of essential hyperbolic Coxeter polytope as a polytope which fits some minimality conditions. The problem of classification of hyperbolic reflection groups can be easily reduced to classification of essential Coxeter polytopes. We determine a potentially large combinatorial class of polytopes containing, in particular, all the compact hyperbolic Coxeter polytopes of dimens...

On hyperbolic Coxeter polytopes with mutually intersecting facets

We prove that, apart from some well-known low-dimensional examples, any compact hyperbolic Coxeter polytope has a pair of disjoint facets. This is one of very few known general results concerning combinatorics of compact hyperbolic Coxeter polytopes. We also obtain a similar result for simple non-compact polytopes.

The Volume of Hyperbolic Coxeter Polytopes of Even Dimension

Hyperbolic Coxeter N-polytopes with N + 2 Facets

In this paper, we classify all the hyperbolic non-compact Coxeter polytopes of finite volume combinatorial type of which is either a pyramid over a product of two simplices or a product of two simplices of dimension greater than one. Combined with results of Kaplinskaja [5] and Esselmann [3] this completes the classification of hyperbolic Coxeter n-polytopes of finite volume with n + 2 facets.