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.