Quadratic Integers and Coxeter Groups

Johnson, Norman W. ; Weiss, Asia Ivić

Canadian journal of mathematics, Tome 51 (1999) no. 6, pp. 1307-1336

Résumé

Matrices whose entries belong to certain rings of algebraic integers can be associated with discrete groups of transformations of inversive $n$ -space or hyperbolic $(n+1)-\text{space}\,{{\text{H}}^{n+1}}$ . For small $n$ , these may be Coxeter groups, generated by reflections, or certain subgroups whose generators include direct isometries of ${{\text{H}}^{n+1}}$ . We show how linear fractional transformations over rings of rational and (real or imaginary) quadratic integers are related to the symmetry groups of regular tilings of the hyperbolic plane or 3-space. New light is shed on the properties of the rational modular group $\text{PS}{{\text{L}}_{2}}(\mathbb{Z})$ , the Gaussian modular (Picard) group $\text{PS}{{\text{L}}_{2}}(\mathbb{Z}[i])$ , and the Eisenstein modular group $\text{PS}{{\text{L}}_{2}}(\mathbb{Z}\left[ \omega\right])$ .

Détail

DOI : 10.4153/CJM-1999-060-6

Mots-clés : 11F06, 20F55, 20G20, 20H10, 22E40

Bibliographie

[1] [1] Alperin, R. C., Normal subgroups of PSL(Z[-3]). Proc. Amer.Math. Soc. 124(1996), 2935–2941. Google Scholar

[2] [2] Bianchi, L., Geometrische Darstellung der Gruppen linearer Substitutionenmit ganzen complexen Coefficienten nebst Anwendungen auf die Zahlentheorie. Math. Ann. 38(1891), 313–333; reprinted in Opere vol. I, pt. 1 (Edizione Cremonese, Rome, 1952), 233–258. Google Scholar

[3] [3] Bianchi, L., Sui gruppi de sostituzioni lineari con coeficienti appartenenti a corpi quadratici imaginari. Math. Ann. 40(1892), 332–412; reprinted in Opere vol. I, pt. 1 (Edizione Cremonese, Rome, 1952), 270–373. Google Scholar

[4] [4] Coxeter, H. S. M., Discrete groups generated by reflections. Annals of Math. (2) 35(1934), 588–621; reprinted in Kaleidoscopes: Selected Writings of Coxeter, H. S. M. (eds. Sherk, F. A., P. McMullen, Thompson, A. C., Weiss, A. I.), Wiley, New York, 1995, 145–178. Google Scholar

[5] [5] Coxeter, H. S. M., Regular honeycombs in hyperbolic space. Proc. Internat. Congress of Mathematicians (Amsterdam, 1954), Vol. III, Noordh off, Groningen, and North-Holland, Amsterdam, 1956, 155–169; reprinted in Twelve Geometric Essays, Southern Illinois Univ. Press, Carbondale, IL, and Feffer & Simons, London-Amsterdam, 1968, 199–214. Google Scholar

[6] [6] Coxeter, H. S. M. and Moser, W. O. J., Generators and Relations for Discrete Groups. Ergeb. Math. Grenzgeb. (N. F.), Bd. 14, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957; 4th edn, idem, Berlin-Heidelberg-New York, 1980. Google Scholar

[7] [7] Coxeter, H. S. M. and Whitrow, G. J., World-structure and non-Euclidean honeycombs. Proc. Roy. Soc. London Ser. A 201(1950), 417–437. Google Scholar

[8] [8] Fine, B., Algebraic Theory of the Bianchi Groups. Pure Appl. Math. 129, Marcel Dekker, New York-Basel, 1989. Google Scholar

[9] [9] Fine, B. and Newman, M., The normal subgroup structure of the Picard group. Trans. Amer. Math. Soc. 302(1987), 769–786. Google Scholar

[10] [10] Fricke, R. and Klein, F., Vorlesungen über die Theorie der automorphen Funktionen. Bd. I, Teubner, Leipzig, 1897; 2nd edn, idem, 1926. Google Scholar

[11] [11] Higman, G., Neumann, B. H. and Neumann, H., Embedding theorems for groups. J. London Math. Soc. 24(1949), 247–254. Google Scholar

[12] [12] Hua, L. K., Introduction to Number Theory. Springer-Verlag, Berlin-Heidelberg-New York, 1982 (translated by P. Shiu from the 1957 Chinese original). Google Scholar

[13] [13] Humphreys, J. E., Reflection Groups and Coxeter Groups. Cambridge University Press, Cambridge-New York, 1990, 1992. Google Scholar

[14] [14] Johnson, N. W., Absolute polarities and central inversions. In: The Geometric Vein: The Coxeter Festchrift (eds. Davis, C., Grunbaum, B. and Sherk, F. A.), Springer-Verlag, New York-Heidelberg-Berlin, 1981, 443–464. Google Scholar

[15] [15] Johnson, N. W. and Weiss, A. I., Quaternionic modular groups. Linear Algebra Appl. 295(1999), 159–189. Google Scholar

[16] [16] Klein, F., Ueber die Transformation der elliptischen Functionen und die Auflösung der Gleichungen fünften Grades. Math. Ann. 14(1879), 111–172; reprinted in Gesammelte mathematische Abhandlungen, Bd. III, Springer, Berlin, 1923, 13–75. Google Scholar

[17] [17] Koszul, J. L., Lectures on Hyperbolic Coxeter Groups. Notes by T. Ochiai. University of Notre Dame, Notre Dame, IN, 1968. Google Scholar

[18] [18] Lannér, F., On complexes with transitive groups of automorphisms. Medd. Lunds Univ. Mat. Sem. 11(1950), 1–71. Google Scholar

[19] [19] Mac Lane, S. and Birkhoff, G., Algebra. Macmillan, New York, and Collier-Macmillan, London, 1967; 3rd edn, Chelsea, New York, 1987. Google Scholar

[20] [20] Magnus, W., Noneuclidean Tesselations and Their Groups. Academic Press, New York-London, 1974. Google Scholar

[21] [21] McMullen, P. and Schulte, E., Constructions for regular polytopes. J. Combin. Theory Ser. A 53(1990), 1–28. Google Scholar

[22] [22] Monson, B. and Weiss, A. I., Polytopes related to the Picard group. Linear Algebra Appl. 218(1995), 185–204. Google Scholar

[23] [23] Monson, B. and Weiss, A. I., Eisenstein integers and related C-groups. Geom. Dedicata 66(1997), 99–117. Google Scholar

[24] [24] Neumann, P. M., Stoy, G. A. and Thompson, E. C., Groups and Geometry. Oxford Univ. Press, Oxford-New York-Tokyo, 1994. Google Scholar

[25] [25] Nostrand, B., Schulte, E. and Weiss, A. I., Constructions of chiral polytopes. Congr. Numer. 97(1993), 165–170. Google Scholar

[26] [26] Picard, É., Sur un groupe des transformations des points de l’espace situés du même côté d’un plan. Bull. Soc. Math. France 12(1884), 43–47, reprinted in OEuvres, t. I (Centre National de la Recherche Scientifique, Paris, 1978), 499–504. Google Scholar

[27] [27] Ratcliffe, J. G., Foundations of Hyperbolic Manifolds. Graduate Texts in Math. 149, Springer-Verlag, New York-Berlin-Heidelberg, 1994. Google Scholar

[28] [28] Schlegel, V., Theorie der homogen zusammengesetzen Raumgebilde. Verh. (=Nova Acte) Kaiserl. Leop.-Carol. Deutsch. Akad. Naturforscher 44(1883), 343–459. Google Scholar

[29] [29] Schulte, E. and Weiss, A. I., Chirality and projective linear groups. Discrete Math. 131(1994), 221–261. Google Scholar

[30] [30] Veblen, O. and Young, J. W., Projective Geometry, Vol. I. Ginn, Boston, 1910; idem, 1916; Blaisdell, New York-Toronto-London, 1965. Google Scholar

[31] [31] Veblen, O. and Young, J. W., Projective Geometry, Vol. II. Ginn, Boston, 1918; Blaisdell, New York-Toronto-London, 1965. Google Scholar