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

Abstract

Let S be a subgroup of SL n R, where R is a commutative ring with identity and n^3. The order of S, oS, is the R-ideal generated by x ij , x ii À x jj i j j, where x ij P S. Let E n R be the subgroup of SL n R generated by the elementary matrices. The level of SY lS, is the largest R-ideal q with the property that S contains all the q-elementary matrices and all conjugates of these by elements of E n R. It is clear that lS % oS. Vaserstein has proved that, for all R and for all n^3, the subgroup S is normalized by E n R if and only if lS oS.Let A be an arithmetic Dedekind domain of characteristic zero with only finitely many units. It is known that A Z or A o d , the ring of integers in the imaginary quadratic field Q Àd p , where d is a square-free positive integer. It has been shown that, for all non-zero Z-ideals q, there exist uncountably many normal subgroups of SL 2 Z with order q and level zero. In this paper we extend this result to all but finitely many of the Bianchi groups SL 2 o d . This answers a question of A. Lubotzky.Introduction. Let R be a commutative ring with identity. With each subgroup S of SL n R, where n^2, we associate a pair of R-ideals. The order of SY oS, is the R-ideal generated by x ij Y x ii À x jj i j j, where x ij P S. It is clear that oS f0g if and only if S consists of scalar matrices. Let E n R denote the subgroup of SL n R generated by the elementary matrices. For each R-ideal q let E n RY q be the normal subgroup of E n R generated by the qelementary matrices. The level of SY lS, is the largest ideal q 0 with the property that E n RY q 0 % S. The level is well-defined since E n RY q 2 Á E n RY q 2 E n RY q 1 q 2 . (The concept of level was first introduced by Klein in the 19th century for the modular group SL 2 Z and plays a crucial role, for example, in the theory of congruence subgroups). It is clear that lS % oS. It is known that, when n^3, these ideals can be used to classify the normal subgroups of SL n R. More precisely Vaserstein [13, Theorem 1] has proved that, when n^3, the subgroup S of SL n R is normalized by E n R if and only if oS lS. The relationship between the order and level of a normal subgroup of SL 2 R is however much more complicated. Indeed, as we shall see, this paper shows that for some (important) classes of rings, there is no such relationship.For certain R with ªmanyº units it is known that the order and level of a normal subgroup of SL 2 R are closely connected. Before proceeding to describe these and other results we