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Zbl 0999.30028
Drumm, Todd A.; Poritz, Jonathan A.
Ford and Dirichlet domains for cyclic subgroups of $\text{PSL}_2({\Bbb C})$ acting on ${{\Bbb H}^3_{\Bbb R}}$ and ${\partial}{{\Bbb H}^3_{\Bbb R}}$.
(English)
[J] Conform. Geom. Dyn. 3, No. 8, 116-150, electronic only (1999). ISSN 1088-4173/e

Summary: Let $\Gamma$ be a cyclic subgroup of $\text{PSL}_2({\Bbb C})$ generated by a loxodromic element. The Ford and Dirichlet fundamental domains for the action of $\Gamma$ on ${{\Bbb H}^3_{\Bbb R}}$ are the complements of configurations of half-balls centered on the plane at infinity ${\partial}{{\Bbb H}^3_{\Bbb R}}$. {\it T. Jørgensen} [Math. Scand. 33, 250-260 (1973; Zbl 0286.30017)] proved that the boundary of the intersection of the Ford fundamental domain with ${\partial}{{\Bbb H}^3_{\Bbb R}}$ 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 ${{\Bbb H}^3_{\Bbb R}}$. We give new proofs of Jørgensen's results, prove analogous facts for Dirichlet domains and for Ford and Dirichlet domains in the interior of ${{\Bbb H}^3_{\Bbb R}}$, and give a complete decomposition of the parameter space by the combinatorial type of the corresponding fundamental domain.
MSC 2000:
*30F40 Kleinian groups
20H10 Fuchsian groups and their generalizations (group theory)

Keywords: Ford domain; Dirichlet domain

Citations: Zbl 0286.30017

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