Poincar\'e Bisectors in Hyperbolic Spaces

TL;DR

This paper derives explicit formulas for bisectors in hyperbolic spaces, compares them with isometric spheres, and applies these results to construct fundamental domains and find generators for various discrete groups.

Contribution

It provides new explicit formulas for hyperbolic bisectors and introduces an implementable algorithm for constructing fundamental domains and generators.

Findings

Explicit formulas for hyperbolic bisectors in 2D and 3D.

Comparison with isometric spheres in Ford domains.

Algorithm DAFC for finding generators of discrete groups.

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.