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Almost all Bianchi groups have free, non-cyclic quotients

  • A. W. Mason (a1), R. W. K. Odoni (a1) and W. W. Stothers (a1)
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Let d be a square-free positive integer and let O (= Od) be the ring of integers of the imaginary quadratic number field ℚ(-d). The groups PSL2(O) are called the Bianchi groups after Luigi Bianchi who made the first important contribution 1 to their study in 1892. Since then they have attracted considerable attention particularly during the last thirty years. Their importance stems primarily from their action as discrete groups of isometries on hyperbolic 3-space, H3. As a consequence they play an important role in hyperbolic geometry, low-dimensional topology together with the theory of discontinuous groups and automorphic forms. In addition they are of particular significance in the class of linear groups over Dedekind rings of arithmetic type. Serre9 has proved that in this class the Bianchi groups (along with, for example, the modular group, PSL2(z), where z is the ring of rational integers) have an exceptionally complicated (non-congruence) subgroup structure.

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1Bianchi, L.. Sui gruppi de sostituzioni lineari con coefficienti appartenenti a corpi quadratici immaginari. Math. Ann. 40 (1892), 332412.
2Burgess, D. A.. On character sums and L-series II. Proc. London Math. Soc. (3) 13 (1963), 524536.
3Fine, B.. Algebraic Theory of the Bianchi Groups (Marcel Dekker, 1989).
4Flge, D.. Zur Struktur der PSL 2 ber einigen imaginr-quadratischen Zahlringen. Math. Z. 183 (1983), 255279.
5Geunewald, F. J. and Schwermek, J.. Free non-abelian quotients of SL 2 over orders of imaginary quadratic number fields. J. Algebra 69 (1981), 298304.
6Hua, L. K.. Introduction to Number Theory (Springer-Verlag, 1982).
7Neumann, P. M.. The SQ-universality of some finitely presented groups. J. Austral. Math. Soc. 16 (1973), 16.
8Riley, R.. Applications of a computer implementation of Poincare's theorem on fundamental polyhedra. Math. Comp. 40 (1983), 607632.
9Serre, J.-P.. La problme des groupes de congruence pour SL 2. Ann. of Math. 92 (1970), 489527.
10Swan, R. G.. Generators and relations for certain special linear groups. Adv. in Math. 6 (1971), 177.
11Zimmert, R.. Zur SL 2 der ganzen Zahlen eines imaginr-quadratischen Zahlkrpers. Invent. Math. 19 (1973), 7381.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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