Residual properties of free pro-p groups

Recall that if S is a class of groups, then a group G is residually-S if, f or any element 1 not equal g is an element of G, there is a normal subgroup N of G such that g is not an element of N and G/N is an element of S. Let A be a commutative Noetherian local pro-p ring, with a maximal ideal M. Rec all that the first congruence subgroup of SLd(A) is: SLd1 (A) = ker (SLd(La mbda) --> SLd(Lambda /M)). Let K subset of or equal to N. We define S-Lambda(K) = boolean OR (d is an element ofK){open subgroups of SLd1(A)}. We show that if K is infinite, the n for Lambda = F-p[[t]] and for Lambda = Z(p) a finitely generated non-abel ian free pro-p group is residually-S-Lambda(K). We apply a probabilistic me thod, combined with Lie methods and a result on random generation in simple algebraic groups over local fields. It is surprising that the case of zero characteristic is deduced from the positive characteristic case.