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

Abstract

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.

The cusp amplitudes and quasi-level of a congruence subgroup of SL2 over any Dedekind domain

“…Corollary 5.7 is stronger than Theorem 5.2 if n is sufficiently small compared to d. Examples 4.10 and 5.4 show that Condition L cannot simply be dropped in Theorem 5.6 and Corollary 5.7. But for arithmetic Dedekind domains D, one can prove somewhat weaker results than Theorem 5.6 and Corollary 5.7 for congruence subgroups that do not satisfy Condition L. The key is that by results of Mason [9] one can still control the relation between l(N ) and ql(N ) (see our Remark 4.11(b)). We content ourselves with the function field case.…”

“…We only give the proof for characteristic 2 as they are almost the same. By Mason [9,Theorem 3.14] (cf. also the end of Section 3 in [9]), we have s 2 (o(N )) 2 l(N ), where s is the product of all prime ideals p in D with |D/p| = 2.…”

Normal Subgroups of Prescribed Order and Zero Level of the Modular Group and Related Groups

“…When q # k[x] the proof is based on previous results [6] of the first author. For the case when q = k[x] the proof makes use of a special automorphism of SL t ( (2, A) it can be shown [9] that the order and level of an element of 3(2, A) are closely related. In particular it is known [7, Theorem 2.2] that 3(2, A) = £f(2, A), when 6sA*.…”

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

“…It is known [6] that n 0 A Y. Moreover by [6, Theorems 3.6, 3.10, 3.14] there is a function 0, defined on the Aideals with the property that, if N is a normal, non-central subgroup of SL 2 A, then 0q % lN, where q oN.…”

“…Moreover by [6, Theorems 3.6, 3.10, 3.14] there is a function 0, defined on the Aideals with the property that, if N is a normal, non-central subgroup of SL 2 A, then 0q % lN, where q oN. (When A is not totally imaginary, for example, 0q 12q, [6,Theorem 3.10].) It follows in particular that, if 6 is a unit in such A, then oN lN.…”