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Ford and Dirichlet domains for cyclic subgroups of ${\text{PSL}}_{2}\left(ℂ\right)$ acting on ${ℍ}_{ℝ}^{3}$ and $\partial {ℍ}_{ℝ}^{3}$. (English) Zbl 0999.30028
Summary: Let ${\Gamma }$ be a cyclic subgroup of ${\text{PSL}}_{2}\left(ℂ\right)$ generated by a loxodromic element. The Ford and Dirichlet fundamental domains for the action of ${\Gamma }$ on ${ℍ}_{ℝ}^{3}$ are the complements of configurations of half-balls centered on the plane at infinity $\partial {ℍ}_{ℝ}^{3}$. T. Jørgensen [Math. Scand. 33, 250-260 (1973; Zbl 0286.30017)] proved that the boundary of the intersection of the Ford fundamental domain with $\partial {ℍ}_{ℝ}^{3}$ always consists of either two, four, or six circular arcs and stated that an arbitrarily large number of hemispheres could contribute faces to the Ford domain in the interior of ${ℍ}_{ℝ}^{3}$. We give new proofs of Jørgensen’s results, prove analogous facts for Dirichlet domains and for Ford and Dirichlet domains in the interior of ${ℍ}_{ℝ}^{3}$, and give a complete decomposition of the parameter space by the combinatorial type of the corresponding fundamental domain.
##### MSC:
 30F40 Kleinian groups 20H10 Fuchsian groups and their generalizations (group theory)
##### Keywords:
Ford domain; Dirichlet domain