Dirichlet-Ford domains and double Dirichlet domains

We continue investigations started by Lakeland on Fuchsian and Kleinian groups which
have a Dirichlet fundamental domain that also is a Ford domain in the upper half-space model
of hyperbolic 2- and 3-space, or which have a Dirichlet domain with multiple centers. Such
domains are called DF-domains and Double Dirichlet domains respectively. Making use of
earlier obtained concrete formulas for the bisectors defining the Dirichlet domain of center
i ∈ H 2 or center j ∈ H 3 , we obtain a simple condition on the matrix entries of the side-
pairing transformations of the fundamental domain of a Fuchsian or Kleinian group to be a
DF-domain. Using the same methods, we also complement a result of Lakeland stating that a
cofinite Fuchsian group has a DF domain (or a Dirichlet domain with multiple centers) if and
only if it is an index 2 subgroup of the discrete group G of reflections in a hyperbolic polygon.