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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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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 PDF
Conform. Geom. Dyn. 3 (1999), 116-150 Request permission

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
  • 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