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Conformal Geometry and Dynamics
Conformal Geometry and Dynamics
ISSN 1088-4173


Ford and Dirichlet domains
for cyclic subgroups
of $PSL_2({\mathbb C})$ acting on ${{\mathbb H}^3_{\mathbb R}}$ and ${\partial}{{\mathbb H}^3_{\mathbb R}}$

Authors: Todd A. Drumm and Jonathan A. Poritz
Journal: Conform. Geom. Dyn. 3 (1999), 116-150
MSC (1991): Primary 20H10; Secondary 57M60, 57S30, 57S25
Published electronically: October 25, 1999
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Abstract: Let $\Gamma$ be a cyclic subgroup of $PSL_2({\mathbb C})$ generated by a loxodromic element. The Ford and Dirichlet fundamental domains for the action of $\Gamma$ on ${{\mathbb H}^3_{\mathbb R}}$ are the complements of configurations of half-balls centered on the plane at infinity ${\partial}{{\mathbb H}^3_{\mathbb R}}$. 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 ${\partial}{{\mathbb H}^3_{\mathbb R}}$ 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 ${{\mathbb H}^3_{\mathbb R}}$. We give new proofs of Jørgensen's results, prove analogous facts for Dirichlet domains and for Ford and Dirichlet domains in the interior of ${{\mathbb H}^3_{\mathbb R}}$, and give a complete decomposition of the parameter space by the combinatorial type of the corresponding fundamental domain.

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Additional Information

Todd A. Drumm
Affiliation: Department of Mathematics and Statistics, Swarthmore College, Swarthmore, PA 19081

Jonathan A. Poritz
Affiliation: Department of Mathematics, Georgetown University, Washington, DC 20057

PII: S 1088-4173(99)00042-9
Keywords: Fundamental domain, Ford domain, Dirichlet domain, hyperbolic geometry
Published electronically: October 25, 1999
Additional Notes: The first author was partially supported by the Swarthmore College Research Fund.
The second author was partially supported by NSF grant DMS-9403784.
Article copyright: © Copyright 1999 American Mathematical Society