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

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- by Todd A. Drumm and Jonathan A. Poritz
- Conform. Geom. Dyn.
**3**(1999), 116-150 - DOI: https://doi.org/10.1090/S1088-4173-99-00042-9
- 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.

## References

- Alan F. Beardon,
*The geometry of discrete groups*, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR**698777**, DOI 10.1007/978-1-4612-1146-4 - Troels Jørgensen,
*On cyclic groups of Möbius transformations*, Math. Scand.**33**(1973), 250–260 (1974). MR**348103**, DOI 10.7146/math.scand.a-11487 - Ivan Niven, Herbert S. Zuckerman, and Hugh L. Montgomery,
*An introduction to the theory of numbers*, 5th ed., John Wiley & Sons, Inc., New York, 1991. MR**1083765**

## Bibliographic Information

**Todd A. Drumm**- Affiliation: Department of Mathematics and Statistics, Swarthmore College, Swarthmore, PA 19081
- Email: tad@swarthmore.edu
**Jonathan A. Poritz**- Affiliation: Department of Mathematics, Georgetown University, Washington, DC 20057
- Email: poritz@math.georgetown.edu
- 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. - © Copyright 1999 American Mathematical Society
- Journal: Conform. Geom. Dyn.
**3**(1999), 116-150 - MSC (1991): Primary 20H10; Secondary 57M60, 57S30, 57S25
- DOI: https://doi.org/10.1090/S1088-4173-99-00042-9
- MathSciNet review: 1716572