Ford and Dirichlet Domains for Cyclic Subgroups of Psl 2 ( C ) Acting on H

Let Γ be a cyclic subgroup of PSL2(C) generated by a loxodromic element. The Ford and Dirichlet fundamental domains for the action of Γ on HR are the complements of configurations of half-balls centered on the plane at infinity ∂HR. Jørgensen (On cyclic groups of Möbius transformations, Math. Scand. 33 (1973), 250–260) proved that the boundary of the intersection of the Ford fundamental domain with ∂HR 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 HR. We give new proofs of Jørgensen’s results, prove analogous facts for Dirichlet domains and for Ford and Dirichlet domains in the interior of HR, and give a complete decomposition of the parameter space by the combinatorial type of the corresponding fundamental domain.