Poincare ́ Bisectors in Hyperbolic Spaces∗

We determine explicit formulas for the Poincare ́ bisectors used in constructing a Dirichlet fundamental domain in hyperbolic two and three space. They are compared with the isometric spheres in the upper half space and plane models of hyperbolic space. This is used to revisit classical results on the Bianchi groups and also some recent results on groups having either a Dirichlet domain with at least two centers or a Dirichlet domain which is also a Ford domain. It is also shown that the relative position of the Poincare ́ bisector of an isometry and its inverse, determine whether the isometry is elliptic, parabolic or hyperbolic.