RESIDUAL PROPERTIES OF FREE PRO-P GROUPS

Abstract

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 SL_d(Λ) is: SL¹_d(Λ) = ker (SL_d(Λ) → SL_d(Λ/M)).

Let K ⊆ ℕ. We define S_Λ(K) = ∪_d∈K{open subgroups of SL¹_d(Λ)}. We show that if K is infinite, then for Λ = [ ]_p[[t]] and for Λ = ℤ_p a finitely generated non-abelian free pro-p group is residually-S_Λ(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.