Properties of Coxeter Andreev’s Tetrahedrons

Geometrization of 3-dimensional Coxeter group manifolds and Singer’s conjecture

Associated to any Coxeter system (W,S), there is a labeled simplicial complex L and a contractible CW-complex ΣL (the Davis complex) on which W acts properly and cocompactly. ΣL admits a cellulation under which the nerve of each vertex is L. It follows that if L is a triangulation of S , then ΣL is a contractible n-manifold. We prove that a special case of the Singer conjecture for ΣL (on the v...

Université De Provence

In 1970, E. M. Andreev published a classification of all three dimensional compact hyperbolic polyhedra having non-obtuse dihedral angles [3]. Given a combinatorial description of a polyhedron, C, Andreev’s Theorem provides five classes of linear inequalities, depending on C, for the dihedral angles, which are necessary and sufficient conditions for the existence of a hyperbolic polyhedron real...

Constructing Hyperbolic Polyhedra Using Newton's Method

We demonstrate how to construct three-dimensional compact hyperbolic polyhedra using Newton’s Method. Under the restriction that the dihedral angles are non-obtuse, Andreev’s Theorem [5, 6] provides as necessary and sufficient conditions five classes of linear inequalities for the dihedral angles of a compact hyperbolic polyhedron realizing a given combinatorial structure C. Andreev’s Theorem a...

Complete Growth Series of Coxeter Groups with more than Three Generators

On the Derived Length of Coxeter Groups

We characterize certain properties of the derived series of Coxeter groups by properties of the corresponding Coxeter graphs. In particular, we give necessary and sufficient conditions for a Coxeter group to be quasiperfect.