Dedekind Ring Of Arithmetic Type Research Articles

Commutators of Congruence Subgroups in the Arithmetic Case

In a joint paper of the author with Alexei Stepanov, it was established that for any two comaximal ideals A and B of a commutative ring R, A + B = R, and any n ≥ 3 one has [E(n,R,A),E(n,R,B)] = E(n,R,AB). Alec Mason and Wilson Stothers constructed counterexamples demonstrating that the above equality may fail when A and B are not comaximal, even for such nice rings as ℤ [i]. The present note proves a rather striking result that the above equality and, consequently, also the stronger equality [GL(n,R,A), GL(n,R,B)] = E(n,R,AB) hold whenever R is a Dedekind ring of arithmetic type with infinite multiplicative group. The proof is based on elementary calculations in the spirit of the previous papers by Wilberd van der Kallen, Roozbeh Hazrat, Zuhong Zhang, Alexei Stepanov, and the author, and also on an explicit computation of the multirelative SK1 from the author’s paper of 1982, which, in its turn, relied on very deep arithmetical results by Jean-Pierre Serre and Leonid Vaserstein (as corrected by Armin Leutbecher and Bernhard Liehl). Bibliography: 50 titles.

UNITRIANGULAR FACTORIZATION OF TWISTED CHEVALLEY GROUPS

Unitriangular factorization is a presentation of a linear group as a product of unipotent radicals of a Borel subgroup and its opposite. Whether this decomposition is known for Chevalley groups over rings of stable rank 1 and some Dedekind rings of arithmetic type, the case of twisted groups has been studied only over finite fields. In the present paper we give a much simpler proof for twisted groups over finite fields and the field of complex numbers.

Parabolic Subgroups of SO2l Over a Dedekind Ring of Arithmetic Type

Let R be a commutative ring all of whose proper factor rings are finite and such that there exists a unit of infinite order. We show that for a subgroup P in G = SO(2l, R), l ≥ 3, containing the Borel subgroup B, the following alternative holds: either P contains a relative elementary subgroup E I for some ideal I = 0 or H is contained in a proper standard parabolic subgroup. For Dedekind rings of arithmetic type this allows us, under some mild additional assumptions on units, to describe completely the overgroups of B in G. Earlier, similar results for the special linear and symplectiv groups were obtained by A. V. Alexandrov and the second author. The proofs in the present paper follow the same general strategy, but are noticeably harder, from a technical viewpoint. Bibliography: 34 titles.

Parabolic subgroups of SL n and Sp2l over a Dedekind ring of arithmetic type

Let R be a commutative ring all of whose proper factor rings are finite and in which a unit of infinite order exists. It is shown that for a subgroup P in G =SL(n, R), n ≥3,or in G =Sp(2l, R), l ≥2, containing the standard Borel subgroup B, the following alternative holds: either P contains a relative elementary subgroup EI for some ideal I≠0 or H is contained in a proper standard parabolic subgroup. For Dedekind rings of arithmetic type, this allows one, under some mild additional assumptions on units, to completely describe the overgroups of B in G. Bibliography: 30 titles.

Products of involutory matrices over rings

Let αϵGL n A be a matrix over a commutative ring A with 1 such that (det α) 2 = 1. If α is cyclic, it can be written as a product of at most three involutions. When A satisfies the first Bass stable range condition, then α can be written as a product of at most five involutions. If in addition either n ⩽ 3 or n = 4 and det α = −1, then α can be written as a product of at most four involutions. When A is a Dedekind ring of arithmetic type, the number of involutions needed to express α is uniformly bounded for any n ⩾ 3. When A = C[ x] the number of involutions is unbounded for any n ≥ 2.

Congruence hulls in SL n

Let D be a Dedekind ring. For a large class of subgroups S of SL n ( D) (which includes, for example, all subgroups of finite index and all non-central, normal subgroups) it is proved that there is a (unique) smallest congruence subgroup of SL n ( D) containing S. This is called the congruence hull of S and is denoted by C( S). Let A be a Dedekind ring of arithmetic type and let A ∗ denote its units. From well-known results of Bass, Liehl, Milnor, Serre and Vaserstein it follows that, when A ∗ is infinite or n≥3, every subgroup S of SL n ( A) containing a non-central, normal subgroup has a congruence hull. In addition the index | C( S): S| divides m, the number of roots of unity in A. When n = 2 and A ∗ is finite the situation is completely different. Using previous results of the author it is proved that, for any such A and any finitely generated group G, there exists a subgroup S of SL 2( A), normal in its congruence hull, such that G is isomorphic to C( S), modulo S.

Almost all Bianchi groups have free, non-cyclic quotients

Let d be a square-free positive integer and let O (= Od) be the ring of integers of the imaginary quadratic number field ℚ(-d). The groups PSL2(O) are called the Bianchi groups after Luigi Bianchi who made the first important contribution 1 to their study in 1892. Since then they have attracted considerable attention particularly during the last thirty years. Their importance stems primarily from their action as discrete groups of isometries on hyperbolic 3-space, H3. As a consequence they play an important role in hyperbolic geometry, low-dimensional topology together with the theory of discontinuous groups and automorphic forms. In addition they are of particular significance in the class of linear groups over Dedekind rings of arithmetic type. Serre9 has proved that in this class the Bianchi groups (along with, for example, the modular group, PSL2(z), where z is the ring of rational integers) have an exceptionally complicated (non-congruence) subgroup structure.

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

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.

Free quotients of congruence subgroups of SL2 over a Dedekind ring of arithmetic type contained in a function field

Let R be a commutative ring with identity and let q be an ideal in R. For each n ≽ 2, 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. We put SLn(R, q) = Ker(SLn(R)→SLn(R/q)), the principal congruence subgroup of GLn(R) of level q. (By definition En(R, R) = En(R) and SLn(R, R) = SLn(R).)

Improvement of K2-stability under transitive actions of elementary groups

For any ring A with finite stable rank sr(A) the canonical map Kz(n I,A)+K2(A) is known to be surjective, if n r sr(A) + 2 and bijective, if n 2 sr(A) + 3 (cf. [3,4,7,9, 111). Moreover, examples are known where these bounds are sharp (cf. [4]). Thus we have to impose additional assumptions on A to get an improvement of the stability behaviour of Kz. We show that the bounds improve by 1, provided certain actions of elementary groups are transitive. As a consequence we get a result, proved by van der Kallen [5] with a different technique, that the map K2(2,A)*K2(A) is surjective and the map K2(3,A)~Kz(A) is bijective, if A is a Dedekind ring of arithmetic type as in Bass-Milnor-Serre [2] and with infinitely many units. I am indebted to W. van der Kallen, whose very helpfully remarks lead to the final version of this paper.

ON THE GROUPSL2OVER DEDEKIND RINGS OF ARITHMETIC TYPE

It is proved that the group of matrices of order two with determinant 1 over a Dedekind ring of arithmetic type is generated by elementary matrices if there are infinitely many invertible elements in this ring. We also obtain a more general result, describing the group generated by elementary matrices belonging to a congruence subgroup. Bibliography: 6 items.