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Ford and Dirichlet domains for cyclic subgroups of PSL 2 () acting on 3 and 3 . (English) Zbl 0999.30028
Summary: Let Γ be a cyclic subgroup of PSL 2 () generated by a loxodromic element. The Ford and Dirichlet fundamental domains for the action of Γ on 3 are the complements of configurations of half-balls centered on the plane at infinity 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 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 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 3 , and give a complete decomposition of the parameter space by the combinatorial type of the corresponding fundamental domain.
MSC:
30F40Kleinian groups
20H10Fuchsian groups and their generalizations (group theory)