The congruence subgroup problem for finitely generated nilpotent groups

The congruence subgroup problem for a finitely generated group and for G Aut./ asks whether the map OG ! Aut. O / is injective, or more generally, what its kernel C.G; / is. Here OX denotes the profinite completion of X. In the case G D Aut./, we write C./ D C.Aut./; /. Let be a finitely generated group, N D =.; ., and D N =tor. N / . Z.d/. Define (for presented) In this paper we show that, when is nilpotent, there is a canonical isomorphism (for presented) In other words, C./ is completely determined by the solution to the classical congruence subgroup problem for the arithmetic group Aut./. In particular, in the case where D .n;c is a finitely generated free nilpotent group of class c on nelements, we get that C.n;c/ D C.Z.n// D e whenever n 3, and C.2;c/ D C.Z.2// D FO is the free profinite group on countable number of generators.