On superintegral Kleinian sphere packings, bugs, and arithmetic groups

References

[1] M. Belolipetsky, Arithmetic hyperbolic reflection groups, Bull. Amer. Math. Soc. (N. S.) 53 (2016), no. 3, 437–475. 10.1090/bull/1530Search in Google Scholar

[2] M. Belolipetsky and J. Mcleod, Reflective and quasi-reflective Bianchi groups, Transform. Groups 18 (2013), no. 4, 971–994. 10.1007/s00031-013-9245-6Search in Google Scholar

[3] R. Benedetti and C. Petronio, Lectures on hyperbolic geometry, Universitext, Springer, Berlin 1992. 10.1007/978-3-642-58158-8Search in Google Scholar

[4] L. Bianchi, Sui gruppi di sostituzioni lineari con coefficienti appartenenti a corpi quadratici immaginarî, Math. Ann. 40 (1892), no. 3, 332–412. 10.1007/BF01443558Search in Google Scholar

[5] M. Bonk, B. Kleiner and S. Merenkov, Rigidity of Schottky sets, Amer. J. Math. 131 (2009), no. 2, 409–443. 10.1353/ajm.0.0045Search in Google Scholar

[6] A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962), 485–535. 10.2307/1970210Search in Google Scholar

[7] B. H. Bowditch, Geometrical finiteness for hyperbolic groups, J. Funct. Anal. 113 (1993), no. 2, 245–317. 10.1006/jfan.1993.1052Search in Google Scholar

[8] R. Brooks, The spectral geometry of the Apollonian packing, Comm. Pure Appl. Math. 38 (1985), no. 4, 359–366. 10.1002/cpa.3160380402Search in Google Scholar

[9] R. Brooks, Circle packings and co-compact extensions of Kleinian groups, Invent. Math. 86 (1986), no. 3, 461–469. 10.1007/BF01389263Search in Google Scholar

[10] P. G. Doyle, On the bass note of a Schottky group, Acta Math. 160 (1988), no. 3–4, 249–284. 10.1007/BF02392277Search in Google Scholar

[11] F. Esselmann, Über die maximale Dimension von Lorentz-Gittern mitcoendlicher Spiegelungsgruppe, J. Number Theory 61 (1996), no. 1, 103–144. 10.1006/jnth.1996.0141Search in Google Scholar

[12] R. Frigerio, Commensurability of hyperbolic manifolds with geodesic boundary, Geom. Dedicata 118 (2006), 105–131. 10.1007/s10711-005-9028-xSearch in Google Scholar

[13] B. Iversen, Hyperbolic geometry, London Math. Soc. Stud. Texts 25, Cambridge University, Cambridge 1992. 10.1017/CBO9780511569333Search in Google Scholar

[14] M. Kapovich, Hyperbolic manifolds and discrete groups, Progr. Math. 183, Birkhäuser, Boston 2001. Search in Google Scholar

[15] A. Kontorovich, From Apollonius to Zaremba: Local-global phenomena in thin orbits, Bull. Amer. Math. Soc. (N. S.) 50 (2013), no. 2, 187–228. 10.1090/S0273-0979-2013-01402-2Search in Google Scholar

[16] A. Kontorovich, Letter to Bill Duke, 2017, available at http://sites.math.rutgers.edu/~alexk/files/LetterToDuke.pdf. Search in Google Scholar

[17] A. Kontorovich and K. Nakamura, Geometry and arithmetic of crystallographic sphere packings, Proc. Natl. Acad. Sci. USA 116 (2019), no. 2, 436–441. 10.1073/pnas.1721104116Search in Google Scholar PubMed PubMed Central

[18] J. C. Lagarias, C. L. Mallows and A. R. Wilks, Beyond the Descartes circle theorem, Amer. Math. Monthly 109 (2002), no. 4, 338–361. 10.1080/00029890.2002.11920896Search in Google Scholar

[19] D. Martin, Continued fractions in non-Euclidean imaginary quadratic fields, preprint (2019), https://arxiv.org/abs/1908.00121. Search in Google Scholar

[20] B. Maskit, Kleinian groups, Grundlehren Math. Wiss. 287, Springer, Berlin 1988. 10.1007/978-3-642-61590-0Search in Google Scholar

[21] J. A. McLeod, Arithmetic hyperbolic reflection groups, Ph.D. thesis, University of Durham Thesis, 2013. Search in Google Scholar

[22] J. J. Millson, On the first Betti number of a constant negatively curved manifold, Ann. of Math. (2) 104 (1976), no. 2, 235–247. 10.2307/1971046Search in Google Scholar

[23] D. W. Morris, Introduction to arithmetic groups, Deductive Press, 2015. Search in Google Scholar

[24] P. J. Nicholls, The ergodic theory of discrete groups, London Math. Soc. Lecture Note Ser. 143, Cambridge University, Cambridge 1989. 10.1017/CBO9780511600678Search in Google Scholar

[25] S. J. Patterson, The limit set of a Fuchsian group, Acta Math. 136 (1976), no. 3–4, 241–273. 10.1007/BF02392046Search in Google Scholar

[26] R. S. Phillips and P. Sarnak, The Laplacian for domains in hyperbolic space and limit sets of Kleinian groups, Acta Math. 155 (1985), no. 3–4, 173–241. 10.1007/BF02392542Search in Google Scholar

[27] J. G. Ratcliffe, Foundations of hyperbolic manifolds, 2nd ed., Grad. Texts in Math. 149, Springer, New York, 2006. Search in Google Scholar

[28] T. Roblin, Un théorème de Fatou pour les densités conformes avec applications aux revêtements galoisiens en courbure négative, Israel J. Math. 147 (2005), 333–357. 10.1007/BF02785371Search in Google Scholar

[29] A. Selberg, On discontinuous groups in higher-dimensional symmetric spaces, Contributions to function theory, Tata Institute of Fundamental Research, Bombay (1960), 147–164. Search in Google Scholar

[30] K. E. Stange, The Apollonian structure of Bianchi groups, Trans. Amer. Math. Soc. 370 (2018), no. 9, 6169–6219. 10.1090/tran/7111Search in Google Scholar

[31] D. Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math. 153 (1984), no. 3–4, 259–277. 10.1007/BF02392379Search in Google Scholar

[32] W. Thurston, Geometry and topology of 3-manifolds, Princeton Lecture Notes, 1981. Search in Google Scholar

[33] P. Tukia, The Hausdorff dimension of the limit set of a geometrically finite Kleinian group, Acta Math. 152 (1984), no. 1–2, 127–140. 10.1007/BF02392194Search in Google Scholar

[34] E. B. Vinberg, The groups of units of certain quadratic forms, Mat. Sb. (N. S.) 87(129) (1972), 18–36. Search in Google Scholar

[35] E. B. Vinberg and O. V. Shvartsman, Discrete groups of motions of spaces of constant curvature, Geometry. II, Encyclopaedia Math. Sci. 29, Springer, Berlin (1993), 139–248. 10.1007/978-3-662-02901-5_2Search in Google Scholar

[36] J. B. Wilker, Inversive geometry, The geometric vein, Springer, New York (1981), 379–442. 10.1007/978-1-4612-5648-9_27Search in Google Scholar