The Apollonian structure of Bianchi groups
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- by Katherine E. Stange PDF
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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.References
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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