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 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.
Cited by
1 articles.
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1. Residual properties of free groups and probabilistic methods;Journal für die reine und angewandte Mathematik (Crelles Journal);2003-01-18