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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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