Recall that if S is a class of groups, then a group G is residually-S if,
for any element 1 ≠ g ∈ G, there is a normal subgroup
N of G such that g ∉ N and G/N ∈ S.
Let Λ be a commutative Noetherian local pro-p ring, with a maximal ideal M.
Recall that the first congruence subgroup of SLd(Λ) is:
SL1d(Λ)
= ker (SLd(Λ) → SLd(Λ/M)).Let K ⊆ ℕ. We define SΛ(K)
= ∪d∈K{open subgroups of SL1d(Λ)}.
We show that if K is infinite, then for Λ = [ ]p[[t]]
and for Λ = ℤp a finitely generated non-abelian free pro-p group is
residually-SΛ(K). We
apply a probabilistic method, 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.
Random generation in semisimple algebraic groups over local fields