Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-26T11:37:38.194Z Has data issue: false hasContentIssue false
  • English
  • Français

Almost all Bianchi groups have free, non-cyclic quotients

Published online by Cambridge University Press:  24 October 2008

A. W. Mason ,
R. W. K. Odoni  and
W. W. Stothers
A. W. Mason
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW
R. W. K. Odoni
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW
W. W. Stothers
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW
Get access Rights & Permissions [Opens in a new window]

Extract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 111 , Issue 1 , January 1992 , pp. 1 - 6
Copyright
Copyright © Cambridge Philosophical Society 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1Bianchi, L.. Sui gruppi de sostituzioni lineari con coefficienti appartenenti a corpi quadratici immaginari. Math. Ann. 40 (1892), 332412.CrossRefGoogle Scholar
2Burgess, D. A.. On character sums and L-series II. Proc. London Math. Soc. (3) 13 (1963), 524536.CrossRefGoogle Scholar
3Fine, B.. Algebraic Theory of the Bianchi Groups (Marcel Dekker, 1989).Google Scholar
4Flge, D.. Zur Struktur der PSL 2 ber einigen imaginr-quadratischen Zahlringen. Math. Z. 183 (1983), 255279.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
6Hua, L. K.. Introduction to Number Theory (Springer-Verlag, 1982).Google Scholar
7Neumann, P. M.. The SQ-universality of some finitely presented groups. J. Austral. Math. Soc. 16 (1973), 16.CrossRefGoogle Scholar
8Riley, R.. Applications of a computer implementation of Poincare's theorem on fundamental polyhedra. Math. Comp. 40 (1983), 607632.Google Scholar
9Serre, J.-P.. La problme des groupes de congruence pour SL 2. Ann. of Math. 92 (1970), 489527.CrossRefGoogle Scholar
10Swan, R. G.. Generators and relations for certain special linear groups. Adv. in Math. 6 (1971), 177.CrossRefGoogle Scholar
11Zimmert, R.. Zur SL 2 der ganzen Zahlen eines imaginr-quadratischen Zahlkrpers. Invent. Math. 19 (1973), 7381.CrossRefGoogle Scholar