Non-Standard, Normal Subgroups and Non-Normal, Standard Subgroups of the Modular Group

Mason, A. W.

doi:10.4153/cmb-1989-016-5

Cited by 8 publications

(2 citation statements)

References 12 publications

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“…It is known ( [12,Theorems 3.2,3.3] and [13]) that both Sf"(2, A) and g o (2, A) are uncountable when A = Z or A = C. (Identical results [12,Theorem 3.2,3.3] also hold for Dedekind rings of similar type to C which are determined by projective curves over any field. This includes the case k [x], where k is any field.)…”

Section: (T) Sl 2 (T)])mentioning

confidence: 98%

The order and level of a subgroup of GL₂ over a Dedekind ring of arithmetic type

Mason

1991

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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SynopsisLet R be a commutative ring and let q be an R-ideal. Let En(R) be the subgroup of GLn(R) generated by the elementary matrices and let En(R, q) be the normal subgroup of En(R) generated by the q-elementary matrices. For each subgroup S of GLn(R) the order of S, o(S), is the R-ideal generated by xij, xii − xjj (i ≠ j), where (xij) ∈ S, and the level of S, l(S), is the largest R-ideal q0 with the property that En (R, q0) ≦ S. It is known that when n ≧ 3, the subgroup S is normalised by En(R) if and only if o(S) = l(S). It is also known that this result does not hold when n = 2. For example, there are uncountably many normal subgroups S of SL2(ℤ) such that o(S) ≠ {0} and l(S) = {0}, where ℤ is the ring of integers. In this paper we prove that, when A is a Dedekind ring of arithmetic type containing infinitely many units, the order q and level q′ of a subgroup S of GL2(A), normalised by E2(A), are closely related. It is proved that Ψ(q)≦q′, where ≦(q) = 12uq, with u the A-ideal generated by u2 − 1 (u ∈ A*), when A is contained in a number field, and Ψ(q) = q3, when A is contained in a function field.

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Section: (T) Sl 2 (T)])mentioning

confidence: 98%

The order and level of a subgroup of GL₂ over a Dedekind ring of arithmetic type

Mason

1991

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…By a theorem originally due to Dirichlet^4 has (at most) finitely many units if and only if A = Z, A -0, for some d, or A = %> = %>(C,P, k), the coordinate ring of an affine curve obtained by removing a (closed) point P from a projective curve C over a finite field k. Although SL 2 (Z) has no free quotient (indeed no torsion-free quotient since it is generated by elements of finite order) it is known that, for all but finitely many Z-ideals q, the group SL 2 (Z,q)/NE 2 (Z,q) has a free quotient (see [14]). Not every SL 2 (%>) has a free quotient.…”

mentioning

confidence: 99%

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

Mason

Odoni

1994

Math. Proc. Camb. Phil. Soc.

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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.

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Free quotients of congruence subgroups of Bianchi groups

Mason

1992

Math. Ann.

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Non-Standard, Normal Subgroups and Non-Normal, Standard Subgroups of the Modular Group

Cited by 8 publications

References 12 publications

The order and level of a subgroup of GL₂ over a Dedekind ring of arithmetic type

The order and level of a subgroup of GL₂ over a Dedekind ring of arithmetic type

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

Free quotients of congruence subgroups of Bianchi groups

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Non-Standard, Normal Subgroups and Non-Normal, Standard Subgroups of the Modular Group

Cited by 8 publications

References 12 publications

The order and level of a subgroup of GL2 over a Dedekind ring of arithmetic type

The order and level of a subgroup of GL2 over a Dedekind ring of arithmetic type

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

Free quotients of congruence subgroups of Bianchi groups

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The order and level of a subgroup of GL₂ over a Dedekind ring of arithmetic type

The order and level of a subgroup of GL₂ over a Dedekind ring of arithmetic type