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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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The Apollonian structure of Bianchi groups
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by Katherine E. Stange PDF
Trans. Amer. Math. Soc. 370 (2018), 6169-6219 Request permission

Abstract:

We study the orbit of $\widehat {\mathbb {R}}$ under the Möbius action of the Bianchi group $\rm {PSL}_2(\mathcal {O}_K)$ on $\widehat {\mathbb {C}}$, where $\mathcal {O}_K$ is the ring of integers of an imaginary quadratic field $K$. The orbit ${\mathcal {S}}_K$, called a Schmidt arrangement, is a geometric realisation, as an intricate circle packing, of the arithmetic of $K$. We give a simple geometric characterisation of certain subsets of ${\mathcal {S}}_K$ generalizing Apollonian circle packings, and show that ${\mathcal {S}}_K$, considered with orientations, is a disjoint union of all primitive integral such $K$-Apollonian packings. These packings are described by a new class of thin groups of arithmetic interest called $K$-Apollonian groups. We make a conjecture on the curvatures of these packings, generalizing the local-to-global conjecture for Apollonian circle packings.
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Additional Information
  • Katherine E. Stange
  • Affiliation: Department of Mathematics, University of Colorado, Campux Box 395, Boulder, Colorado 80309-0395
  • MR Author ID: 845009
  • Email: kstange@math.colorado.edu
  • Received by editor(s): August 4, 2016
  • Received by editor(s) in revised form: October 27, 2016
  • Published electronically: February 8, 2018
  • Additional Notes: The author’s work was sponsored by the National Security Agency under Grants H98230-14-1-0106 and H98230-16-1-0040. The United States goverment is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein.
  • © Copyright 2018 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 370 (2018), 6169-6219
  • MSC (2010): Primary 52C26, 20G30, 11F06, 11R11, 11E57; Secondary 20E08, 20F65, 51F25, 11E39, 11E16
  • DOI: https://doi.org/10.1090/tran/7111
  • MathSciNet review: 3814328