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Dirichlet-Ford Domains and Arithmetic Reflection Groups

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.