Dirichlet-Ford Domains and Arithmetic Reflection Groups

TL;DR

This paper characterizes when Fuchsian groups have Ford domains that are also Dirichlet domains, linking this property to being an index 2 subgroup of a reflection group, and explores related results for Kleinian groups.

Contribution

It establishes a precise criterion for Fuchsian groups to have Ford domains that are also Dirichlet domains, and provides examples of maximal arithmetic hyperbolic reflection groups that are non-congruence.

Findings

Fuchsian groups with Ford and Dirichlet domains are exactly index 2 subgroups of reflection groups.

Constructs an example of a maximal arithmetic hyperbolic reflection group that is not congruence.

Extends results and provides counterexamples for Kleinian groups.

Abstract

In this paper, it is shown that a Fuchsian group, acting on the upper half-plane model for $\mathbb{H}^2$, admits a Ford domain which is also a Dirichlet domain, for some center, if and only if it is an index 2 subgroup of a reflection group. This is used to exhibit an example of a maximal arithmetic hyperbolic reflection group which is not congruence. Analogous results, and counterexamples, are given in the case of Kleinian groups.