Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

The Apollonian structure of Bianchi groups


Author: Katherine E. Stange
Journal: Trans. Amer. Math. Soc.
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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] Christopher J. Bishop and Peter W. Jones, Hausdorff dimension and Kleinian groups, Acta Math. 179 (1997), no. 1, 1-39. MR 1484767, https://doi.org/10.1007/BF02392718
  • [2] Jean Bourgain and Elena Fuchs, A proof of the positive density conjecture for integer Apollonian circle packings, J. Amer. Math. Soc. 24 (2011), no. 4, 945-967. MR 2813334, https://doi.org/10.1090/S0894-0347-2011-00707-8
  • [3] Jean Bourgain and Alex Kontorovich, On the local-global conjecture for integral Apollonian gaskets, Invent. Math. 196 (2014), no. 3, 589-650. With an appendix by Péter P. Varjú. MR 3211042, https://doi.org/10.1007/s00222-013-0475-y
  • [4] Steve Butler, Ron Graham, Gerhard Guettler, and Colin Mallows, Irreducible Apollonian configurations and packings, Discrete Comput. Geom. 44 (2010), no. 3, 487-507. MR 2679050, https://doi.org/10.1007/s00454-009-9216-9
  • [5] John H. Conway, The sensual (quadratic) form, Carus Mathematical Monographs, vol. 26, Mathematical Association of America, Washington, DC, 1997. With the assistance of Francis Y. C. Fung. MR 1478672
  • [6] Leonard Eugene Dickson, History of the theory of numbers. Vol. III: Quadratic and higher forms., With a chapter on the class number by G. H. Cresse, Chelsea Publishing Co., New York, 1966. MR 0245501
  • [7] Joan L. Dyer, Automorphism sequences of integer unimodular groups, Illinois J. Math. 22 (1978), no. 1, 1-30. MR 0460483
  • [8] J. Elstrodt, F. Grunewald, and J. Mennicke, Groups acting on hyperbolic space, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. Harmonic analysis and number theory. MR 1483315
  • [9] Elena Fuchs, Strong approximation in the Apollonian group, J. Number Theory 131 (2011), no. 12, 2282-2302. MR 2832824, https://doi.org/10.1016/j.jnt.2011.05.010
  • [10] Elena Fuchs, Counting problems in Apollonian packings, Bull. Amer. Math. Soc. (N.S.) 50 (2013), no. 2, 229-266. MR 3020827, https://doi.org/10.1090/S0273-0979-2013-01401-0
  • [11] Elena Fuchs and Katherine Sanden, Some experiments with integral Apollonian circle packings, Exp. Math. 20 (2011), no. 4, 380-399. MR 2859897, https://doi.org/10.1080/10586458.2011.565255
  • [12] Ronald L. Graham, Jeffrey C. Lagarias, Colin L. Mallows, Allan R. Wilks, and Catherine H. Yan, Apollonian circle packings: number theory, J. Number Theory 100 (2003), no. 1, 1-45. MR 1971245, https://doi.org/10.1016/S0022-314X(03)00015-5
  • [13] Ronald L. Graham, Jeffrey C. Lagarias, Colin L. Mallows, Allan R. Wilks, and Catherine H. Yan, Apollonian circle packings: geometry and group theory. I. The Apollonian group, Discrete Comput. Geom. 34 (2005), no. 4, 547-585. MR 2173929, https://doi.org/10.1007/s00454-005-1196-9
  • [14] Ronald L. Graham, Jeffrey C. Lagarias, Colin L. Mallows, Allan R. Wilks, and Catherine H. Yan, Apollonian circle packings: geometry and group theory. II. Super-Apollonian group and integral packings, Discrete Comput. Geom. 35 (2006), no. 1, 1-36. MR 2183489, https://doi.org/10.1007/s00454-005-1195-x
  • [15] Gerhard Guettler and Colin Mallows, A generalization of Apollonian packing of circles, J. Comb. 1 (2010), no. 1, [ISSN 1097-959X on cover], 1-27. MR 2675919, https://doi.org/10.4310/JOC.2010.v1.n1.a1
  • [16] Andrew Jensen, Cherry Ng, Evan Oliver, Tyler Schrock, and Katherine E. Stange,
    The Schmidt arrangement of the Eisenstein integers and Apollonian-like packings arising from congruence subgroups,
    In preparation.
  • [17] Gareth A. Jones and John S. Thornton, Automorphisms and congruence subgroups of the extended modular group, J. London Math. Soc. (2) 34 (1986), no. 1, 26-40. MR 859146, https://doi.org/10.1112/jlms/s2-34.1.26
  • [18] Vladimir V. Kisil, Starting with the group $ {\rm SL}_2({\bf R})$, Notices Amer. Math. Soc. 54 (2007), no. 11, 1458-1465. MR 2361159
  • [19] Jerzy Kocik,
    A theorem on circle configuarations, 2007.
    arXiv:0706.0372.
  • [20] Alex Kontorovich,
    The local-global principle for integral soddy sphere packings,
    arXiv:1208.5441.
  • [21] Alex Kontorovich, From Apollonius to Zaremba: local-global phenomena in thin orbits, Bull. Amer. Math. Soc. (N.S.) 50 (2013), no. 2, 187-228. MR 3020826, https://doi.org/10.1090/S0273-0979-2013-01402-2
  • [22] Jeffrey C. Lagarias, Colin L. Mallows, and Allan R. Wilks, Beyond the Descartes circle theorem, Amer. Math. Monthly 109 (2002), no. 4, 338-361. MR 1903421, https://doi.org/10.2307/2695498
  • [23] D. W. Lewis, The isometry classification of Hermitian forms over division algebras, Linear Algebra Appl. 43 (1982), 245-272. MR 656449, https://doi.org/10.1016/0024-3795(82)90258-0
  • [24] David Mumford, Caroline Series, and David Wright, Indra's pearls, Cambridge University Press, Cambridge, 2015. The vision of Felix Klein; With cartoons by Larry Gonick; Paperback edition with corrections; For the 2002 edition see [ MR1913879]. MR 3558870
  • [25] David Mumford, Caroline Series, and David Wright, Indra's pearls, Cambridge University Press, New York, 2002. The vision of Felix Klein. MR 1913879
  • [26] Kei Nakamura,
    The local-global principle for integral bends in orthoplicial apollonian sphere packings,
    arXiv:1401.2980.
  • [27] Morris Newman, Classification of normal subgroups of the modular group, Trans. Amer. Math. Soc. 126 (1967), 267-277. MR 0204375, https://doi.org/10.2307/1994453
  • [28] Bogdan Nica, The unreasonable slightness of $ {\rm E}_2$ over imaginary quadratic rings, Amer. Math. Monthly 118 (2011), no. 5, 455-462. MR 2805033, https://doi.org/10.4169/amer.math.monthly.118.05.455
  • [29] Hee Oh, Apollonian circle packings: dynamics and number theory, Jpn. J. Math. 9 (2014), no. 1, 69-97. MR 3173439, https://doi.org/10.1007/s11537-014-1384-6
  • [30] Dan Romik, The dynamics of Pythagorean triples, Trans. Amer. Math. Soc. 360 (2008), no. 11, 6045-6064. MR 2425702, https://doi.org/10.1090/S0002-9947-08-04467-X
  • [31] Recep Sahin, Sebahattin Ikikardes, and Özden Koruoğlu, Some normal subgroups of the extended Hecke groups $ \overline H(\lambda_p)$, Rocky Mountain J. Math. 36 (2006), no. 3, 1033-1048. MR 2254377, https://doi.org/10.1216/rmjm/1181069444
  • [32] Peter Sarnak, Integral Apollonian packings, Amer. Math. Monthly 118 (2011), no. 4, 291-306. MR 2800340, https://doi.org/10.4169/amer.math.monthly.118.04.291
  • [33] Asmus L. Schmidt, Farey triangles and Farey quadrangles in the complex plane, Math. Scand. 21 (1967), 241-295 (1969). MR 0245525, https://doi.org/10.7146/math.scand.a-10863
  • [34] Asmus L. Schmidt, Diophantine approximation of complex numbers, Acta Math. 134 (1975), 1-85. MR 0422168, https://doi.org/10.1007/BF02392098
  • [35] Asmus L. Schmidt, Diophantine approximation in the field $ {\bf Q}(i(11^{1/2}))$, J. Number Theory 10 (1978), no. 2, 151-176. MR 0485715, https://doi.org/10.1016/0022-314X(78)90033-1
  • [36] Asmus L. Schmidt, Diophantine approximation in the Eisensteinian field, J. Number Theory 16 (1983), no. 2, 169-204. MR 698164, https://doi.org/10.1016/0022-314X(83)90040-9
  • [37] Asmus L. Schmidt, Diophantine approximation in the field $ \mathbb{Q}(i\sqrt2)$, J. Number Theory 131 (2011), no. 10, 1983-2012. MR 2811562, https://doi.org/10.1016/j.jnt.2011.04.002
  • [38] Jean-Pierre Serre, Trees, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR 607504
  • [39] Arseniy Sheydvasser,
    untitled preprint.
  • [40] Frederick Soddy,
    The bowl of integers and the hexlet,
    Nature, 139:77-79, 1937.
  • [41] Katherine E. Stange, The sensual Apollonian circle packing, Expo. Math. 34 (2016), no. 4, 364-395. MR 3578004, https://doi.org/10.1016/j.exmath.2016.01.001
  • [42] Katherine E. Stange, Visualizing imaginary quadratic fields, CMS Notes 48 (2016), no. 4, 16-17. MR 3559745
  • [43] W.A. Stein et al.,
    Sage Mathematics Software (Version 6.4).
    The Sage Development Team, 2015.
    http://www.sagemath.org.
  • [44] A. Muhammed Uludağ and Hakan Ayral,
    Jimm, a Fundamental Involution, 2015.
    arXiv:1501.03787.
  • [45] Xin Zhang, On the Local-global Principle for Integral Apollonian-3 Circle Packings, ProQuest LLC, Ann Arbor, MI, 2014. Thesis (Ph.D.)-State University of New York at Stony Brook. MR 3347084

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 52C26, 20G30, 11F06, 11R11, 11E57, 20E08, 20F65, 51F25, 11E39, 11E16

Retrieve articles in all journals with MSC (2010): 52C26, 20G30, 11F06, 11R11, 11E57, 20E08, 20F65, 51F25, 11E39, 11E16


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

American Mathematical Society