Generators and relations for certain special linear groups
Richard G. Swan · Bulletin of the American Mathematical Society · 1968
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) 112 ).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"~l] 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.