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Transactions of the American Mathematical Society

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The Apollonian structure of Bianchi groups


Author: Katherine E. Stange
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
Published electronically: February 8, 2018
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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
Email: kstange@math.colorado.edu

DOI: https://doi.org/10.1090/tran/7111
Keywords: Apollonian circle packings, projective linear group, M\"obius transformation, thin groups, Bianchi group, imaginary quadratic fields
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.
Article copyright: © Copyright 2018 American Mathematical Society

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