Generators and relations for certain special linear groups

Several years ago I calculated presentations for several of the groups SL(2, R) where R is the ring of integers of a quadratic imaginary number field K = Q((—in)). The method used was extremely tedious and was never published. Recently, while checking these calculations, I discovered a much simpler approach to the problem which I will outline here. The interest in these calculations is considerably increased by recent results of Serre [ô]. He considers the congruence subgroup problem for the groups SL(2, R) where R is the ring of integers 0 of an algebraic number field (and, more generally for R = e[a"~] where a£0 ) . He obtains the expected results [l] , [5] whenever R has a unit of infinite order. Thus the only exceptions are R~Zand the case which I will consider here. Serre has also shown that all of these cases are true exceptions. The case R = Z is, of course, well known. Hopefully, the calculations outlined here will throw some light on the remaining cases. At present, I have only carried out the calculations for fields K with discriminants D between — 1 and — 24. The length of the calculation increases rapidly with \D\ but the calculation could easily be extended to arbitrarily large values of | JOJ by machine computation. This has not been done at the present time. Full details of the calculations will be published elsewhere. I would like to thank H. Bass for communicating Serre's results to me.