Poincaré Bisectors in Hyperbolic Spaces

Abstract

We determine explicit formulas for the bisectors used in constructing a Dirichlet fundamental domain in hyperbolic two and three space. They are compared with the isometric spheres employed in the construction of a Ford domain and used to find a finite set of generators for discrete groups of finite covolume. Applications are given to Fuchsian groups, Kleinian groups, including the Bianchi groups, and for the construction of a finite set of generators of the unit group of the integral group ring of a finite nilpotent group. An easy implementable algorithm, DAFC, is also given and used in the search for generators of discrete groups.

This paper has been withdrawn by the authors because it was not very clear. We have splitted the results into two different papers making them clearer and better understandable. Those papers may be found on arxiv under the names "From the Poincaré Theorem to generators of the unit group of integral group rings of finite groups" and "Dirichlet-Ford domains and Double Dirichlet domains"