Dirichlet-Ford domains and Double Dirichlet domains

TL;DR

This paper characterizes Fuchsian and Kleinian groups with Dirichlet domains that are also Ford domains or have multiple centers, providing explicit conditions on their side-pairing transformations and linking to reflection groups.

Contribution

It derives a simple criterion for identifying DF-domains in Fuchsian and Kleinian groups using bisector formulas and relates these groups to reflection groups in hyperbolic geometry.

Findings

A simple condition on matrix entries for a Dirichlet domain to be a DF-domain.

Extension of Lakeland's result linking cofinite Fuchsian groups with DF-domains to reflection groups.

Explicit formulas for bisectors in hyperbolic space used to analyze fundamental domains.

Abstract

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 $i \in \HQ^2$ or center $j \in \HQ^3$, 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.