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Archiv der Mathematik

, Volume 75, Issue 6, pp 401–409 | Cite as

Normal subgroups of prescribed order and zero level of subgroups of the Bianchi groups

  • A.W. Mason
  • R.M. Scarth
  • 29 Downloads

Abstract.

Abstract. Let S be a subgroup of SL n (R), where R is a commutative ring with identity and \(n \geqq 3\). The order of S, o(S), is the R-ideal generated by \(x_{ij},\ x_{ii} - x_{jj}\ (i \neq j)\), where \((x_{ij}) \in S\). Let E n (R) be the subgroup of SL n (R) generated by the elementary matrices. The level of S, l(S), is the largest R-ideal \(\frak {q}\) with the property that S contains all the \(\frak {q}\)-elementary matrices and all conjugates of these by elements of E n (R). It is clear that \(l(S) \leqq o(S)\). Vaserstein has proved that, for all R and for all \(n \geqq 3\), the subgroup S is normalized by E n (R) if and only if l(S) = o(S).¶Let A be an arithmetic Dedekind domain of characteristic zero with only finitely many units. It is known that \(A = \Bbb {Z}\) or \(A = {\cal O}_d\), the ring of integers in the imaginary quadratic field \(\Bbb {Q}(\sqrt {- d})\), where d is a square-free positive integer. It has been shown that, for all non-zero \(\Bbb {Z}\)-ideals \(\frak {q}\), there exist uncountably many normal subgroups of \(SL_2(\Bbb {Z})\) with order \(\frak {q}\) and level zero. In this paper we extend this result to all but finitely many of the Bianchi groups \(SL_2({\cal O}_d)\). This answers a question of A. Lubotzky.

Keywords

Positive Integer Normal Subgroup Commutative Ring Characteristic Zero Level Zero 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag, Basel 2000

Authors and Affiliations

  • A.W. Mason
    • 1
  • R.M. Scarth
    • 1
  1. 1.Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland, UKGB

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