Ford and Dirichlet domains

for cyclic subgroups

of acting on and

Authors:
Todd A. Drumm and Jonathan A. Poritz

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

Published electronically:
October 25, 1999

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1716572

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a cyclic subgroup of generated by a loxodromic element. The Ford and Dirichlet fundamental domains for the action of on are the complements of configurations of half-balls centered on the plane at infinity . 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 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 . We give new proofs of Jørgensen's results, prove analogous facts for Dirichlet domains and for Ford and Dirichlet domains in the interior of , and give a complete decomposition of the parameter space by the combinatorial type of the corresponding fundamental domain.

**1.**A. Beardon,*The geometry of discrete groups*, Springer-Verlag, New York, 1983. MR**85d:22026****2.**T. Jørgensen,*On cyclic groups of Möbius transformations*, Math. Scand.**33**(1973), 250-260. MR**50:601****3.**I. Niven, H. Zuckerman, and L. Montgomery,*An introduction to the theory of numbers*, John Wiley & Sons, New York, 1991. MR**91i:11001**

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Additional 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

DOI:
https://doi.org/10.1090/S1088-4173-99-00042-9

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