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Free quotients of subgroups of the Bianchi groups whose kernels contain many elementary matrices

Published online by Cambridge University Press:  24 October 2008

A. W. Mason  and
R. W. K. Odoni
A. W. Mason
Affiliation:
University of Glasgow, Department of Mathematics, University Gardens, Glasgow G12 8QW, Scotland
R. W. K. Odoni
Affiliation:
University of Glasgow, Department of Mathematics, University Gardens, Glasgow G12 8QW, Scotland
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Abstract

Let d be a square-free positive integer and let be the ring of integers of the imaginary quadratic number field ℚ(√ − d) The Bianchi groups are the groups SL2() (or PSL2(). Let m be the order of index m in . In this paper we prove that for each d there exist infinitely many m for which SL2(m)/NE2(m) has a free, non-cyclic quotient, where NE2(m) is the normal subgroup of SL2(m) generated by the elementary matrices. When d is not a prime congruent to 3 (mod 4) this result is true for all but finitely many m. The proofs are based on the fundamental paper of Zimmert and its generalization due to Grunewald and Schwermer.

The results are used to extend earlier work of Lubotzky on non-congruence subgroups of SL2(), which involves the concept of the ‘non-congruence crack’. In addition the results highlight a number of low-dimensional anomalies. For example, it is known that [SLn(m), SLnm)] = En(m), when n ≥ 3, where [SLn(m), SLn(m)] is the commutator subgroup of SL(m) and En(m) is the subgroup of SLn(m) generated by the elementary matrices. Our results show that this is not always true when n = 2.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 116 , Issue 2 , September 1994 , pp. 253 - 273
Copyright
Copyright © Cambridge Philosophical Society 1994

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References

REFERENCES

[1]Bass, H.. Algebraic K-theory (Benjamin, 1968).Google Scholar
[2]Bass, H., Milnor, J. and Serre, J. -P., Solution of the congruence subgroup problem for SL n(n ≥ 3) and Sp 2n(n ≥ 2). Publ. Math. I.H.E.S. 33 (1967), 59–137.CrossRefGoogle Scholar
[3]Borevich, Z. I. and Shafarevich, I. R.. Number Theory (Academic Press, 1966).Google Scholar
[4]Cohn, P. M.. On the structure of the GL 2 of a ring. Publ. Math. I.H.E.S. 30 (1966), 365–413.Google Scholar
[5]Cremona, J. E.. On the GL(n) of a Dedekind domain. Quart. J. Math. Oxford (2) 39 (1988), 423–426.CrossRefGoogle Scholar
[6]Dennis, R. K.. The GE 2 property for discrete subrings of ℂ. Proc. Amer. Math. Soc. 50(1975), 77–82.Google Scholar
[7]Fine, B.. Algebraic theory of the Bianchi groups (Marcel Dekker, 1989).Google Scholar
[8]Fine, B. and Newman, M.. The normal subgroup structure of the Picard group. Trans. Amer. Math. Soc. 302 (1987), 769–786.CrossRefGoogle Scholar
[9]Grunewald, F. J. and Schwermer, J.. Free non-abelian quotients of SL 2 over orders of imaginary quadratic number fields. J. Algebra 69 (1981), 298–304.CrossRefGoogle Scholar
[10]Levin, F.. Factor groups of the modular group. J. London Math. Soc. 43 (1968), 195–203.CrossRefGoogle Scholar
[11]Liehl, B.. On the groups SL 2 over orders of arithmetic type. J. reine angew. Math. 323 (1981), 153–171.Google Scholar
[12]Lubotzky, A.. Free quotients and the congruence kernel of SL 2. J. Algebra 77 (1982), 411–418.CrossRefGoogle Scholar
[13]Mason, A. W.. Free quotients of congruence subgroups of SL 2 over a Dedekind ring of arithmetic type contained in a function field. Math. Proc. Cambridge Phil. Soc. 101 (1987), 421–429.Google Scholar
[14]Mason, A. W.. Non-standard, normal subgroups and non-normal, standard subgroups of the modular group. Canad. Math. Bull. 32 (1989), 109–113.CrossRefGoogle Scholar
[15]Mason, A. W.. Free quotients of congruence subgroups of Bianchi groups. Math. Ann. 293 (1992), 65–75.Google Scholar
[16]Mason, A. W., Odoni, R. W. K. and Stothers, W. W.. Almost all Bianchi groups have free, non-cyclic quotients Math. Proc. Cambridge Phil. Soc. 111 (1992), 1–6.Google Scholar
[17]Matsumura, H.. Commutative Algebra (Second Edition) (Benjamin/Cummings, 1980).Google Scholar
[18]Narkiewicz, W.. Elementary and analytic theory of algebraic numbers (PWN-Polish Scientific Publishers, 1974).Google Scholar
[19]Neumann, P. M.. The SQ-universality of some finitely presented groups. J. Australian Math. Soc. 16 (1973), 1–6.Google Scholar
[20]Riley, R.. Applications of a Computer Implementation of Poincaré's Theorem on Fundamental Polyhedra. Math. of Comp. 40 (1983), 607–632.Google Scholar
[21]Serre, J.-P.. Le problème des groupes de congruence pour SL 2. Ann. of Math. 92 (1970), 489–527.Google Scholar
[22]Swan, R. G.. Generators and relations for certain special linear groups. Adv. in Math. 6 (1971), 1–77.Google Scholar
[23]Zimmert, R.. Zur SL 2 der ganzen Zahlen eines imaginär-quadratisehen Zahlkörpers. Invent. Math. 19 (1973), 73–81.Google Scholar