Integers, Modular Groups, and Hyperbolic Space

  • Norman W. Johnson
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 234)


In each of the normed division algebras over the real field \(\mathbb {R}\)—namely, \(\mathbb {R}\) itself, the complex numbers \(\mathbb {C}\), the quaternions IH, and the octonions \(\mathbb {O}\)—certain elements can be characterized as integers. An integer of norm 1 is a unit. In a basic system of integers the units span a 1-, 2-, 4-, or 8-dimensional lattice, the points of which are the vertices of a regular or uniform Euclidean honeycomb. A modular group is a group of linear fractional transformations whose coefficients are integers in some basic system. In the case of the octonions, which have a nonassociative multiplication, such transformations form a modular loop. Each real, complex, or quaternionic modular group can be identified with a subgroup of a Coxeter group operating in hyperbolic space of 2, 3, or 5 dimensions.


  1. 1.
    L.V. Ahlfors, Möbius transformations and Clifford numbers, in Differential Geometry and Complex Analysis, H. E. Rauch memorial volume, ed. by I. Chavel, H. M. Farkas (Springer, Berlin, 1985), pp. 65–73CrossRefGoogle Scholar
  2. 2.
    L. BIanchi, Geometrische Darstellung der Gruppen linearer Substitutionen mit ganzen complexen Coefficienten nebst Anwendungen auf die Zahlentheorie,” Math. Ann.38, 313–333 (1891). Reprinted in Opere, vol. I, pt. 1 (Edizione Cremonese, Rome, 1952), pp. 233–258Google Scholar
  3. 3.
    L. BIanchi, Sui gruppi de sostituzioni lineari con coeficienti appartenenti a corpi quadratici imaginari,” Math. Ann.40, pp. 332–412 (1892). Reprinted in Opere, vol. I, pt. 1 (Edizione Cremonese, Rome, 1952), pp. 270–373Google Scholar
  4. 4.
    P. Boddington, D. Rumynin, On Curtis’ theorem about finite octonionic loops. Proc. Amer. Math. Soc. 135, 1651–1657 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    J.H. Conway, D.A. Smith, On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry (AK Peters, Natick, Mass., 2003)Google Scholar
  6. 6.
    H.S.M. Coxeter, Integral Cayley numbers, Duke Math. J.13, pp. 561–578. Reprinted in Twelve Geometric Essays (Southern Illinois Univ. Press, Carbondale, and Feffer & Simons, London-Amsterdam, 1968) or The Beauty of Geometry: Twelve Essays (Dover, Mineola, N.Y., 1999), pp. 21–39MathSciNetCrossRefGoogle Scholar
  7. 7.
    H.S.M. Coxeter, W.O.J. Moser Generators and Relations for Discrete Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (N.F.), Bd. 14. (Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957; 4th ed., Springer-Verlag, Berlin-Heidelberg-New York, 1980)Google Scholar
  8. 8.
    R.T. Curtis, Construction of a family of Moufang loops. Math. Proc. Cambridge Philos. Soc. 142, 233–237 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    L.E. Dickson, A new simple theory of hypercomplex integers,” J. Math. Pures Appl. (9) 2, pp. 281–326. Reprinted in Collected Mathematical Papers, Vol. VI (Chelsea, Bronx, N.Y., 1983), pp. 531–576Google Scholar
  10. 10.
    L.E. Dickson, Algebras and Their Arithmetics. Univ. of Chicago Press, Chicago, 1923; G.E. Stechert, New York, 1938. Trans. by J.J. Burckhardt, E. Schubarth as Algebren und ihre Zahlentheorie (Orell Füssli, Zurich-Leipzig, 1927)Google Scholar
  11. 11.
    P. Du Val, Homographies, Quaternions, and Rotations (Clarendon Press/Oxford Univ. Press, Oxford, 1964), p. 1964MATHGoogle Scholar
  12. 12.
    N.W. Johnson, Integers. Math. Intelligencer 35, 52–59 (2013). (No. 2)MathSciNetCrossRefGoogle Scholar
  13. 13.
    N.W. Johnson, Geometries and Transformations (Press, Cambridge-New York, Cambridge Univ, 2017). (to appear)MATHGoogle Scholar
  14. 14.
    N.W. Johnson, A.I. Weiss, Quadratic integers and Coxeter groups. Can. J. Math. 51, 1307–1336 (1999a)MathSciNetCrossRefGoogle Scholar
  15. 15.
    N.W. Johnson, A.I. Weiss, Quaternionic modular groups. Linear Algebra Appl. 295, 159–189 (1999b)MathSciNetCrossRefGoogle Scholar
  16. 16.
    R. Lipschitz, Untersuchungen über die Summen von Quadraten (Bonn, M. Cohen, 1886), p. 1886Google Scholar
  17. 17.
    W. Magnus, Noneuclidean Tesselations and Their Groups (Academic Press, New York-London, 1974), p. 1974MATHGoogle Scholar
  18. 18.
    B.R. Monson, A.I. Weiss, Polytopes related to the Picard group. Linear Algebra Appl. 218, 185–204 (1995)MathSciNetCrossRefGoogle Scholar
  19. 19.
    B.R. Monson, A.I. Weiss, Eisenstein integers and related C-groups. Geom. Dedicata. 66, 99–117 (1997)MathSciNetCrossRefGoogle Scholar
  20. 20.
    H. PoincarÉ, Theorie des groupes fuchsiens, Acta Math.1, pp. 1–62. Reprinted in Œuvres, t. II (Gauthier-Villars, Paris, 1952), pp. 108–168. First two sections trans. by J. Stillwell as “Theory of Fuchsian groups” in Sources of Hyperbolic Geometry, History of Mathematics, Vol. 10 (American Math. Soc. and London Math. Soc., Providence, 1996), pp. 123–129 (1882)Google Scholar
  21. 21.
    E. Schulte, A.I. Weiss, Chirality and projective linear groups. Discrete Math. 131, 221–261 (1994)MathSciNetCrossRefGoogle Scholar
  22. 22.
    K.T. Vahlen, Ueber Bewegungen und complexe Zahlen. Math. Ann. 55, 585–593 (1902)MathSciNetCrossRefGoogle Scholar
  23. 23.
    J.B. Wilker, The quaternion formalism for Möbius groups in four or fewer dimensions. Linear Algebra Appl. 190, 99–136 (1993)MathSciNetCrossRefGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Norman W. Johnson
    • 1
  1. 1.NortonUSA

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