NON-NORMAL, STANDARD SUBGROUPS OF THE BIANCHI GROUPS

Let S be a subgroup of SL(n)(K), where K is a Dedekind ring, and let q be the K-ideal generated by x(ij), x(ii) - x(jj) (i not equal j), where (x(ij)) is an element of S. The subgroup S is called standard iff S contains the normal subgroup of SL(n)(K) generated by the q-elementary matrices. It is known that, when n greater than or equal to 3, S is standard iff S is normal in SL,(K). It is also known that every standard subgroup of SL(2)(K) is normal in SL(2)(K) when K is an arithmetic Dedekind domain with infinitely many units. The ring of integers of an imaginary quadratic number field, O, is one example (of three) of such an arithmetic domain with finitely many units. In this paper it is proved that every Bianchi group SL(2)(O) has uncountably many non-normal, standard subgroups. This result is already known for related groups like SL(2)(Z).