Direct products and profinite completions

Abstract

Let G be a finitely generated residually finite group and A a finitely generated normal subgroup. Then G and A are naturally embedded in their respective profinite completions Ĝ and Â. The inclusion A → G induces a morphism (continuous homomorphism) ι :  → Ĝ, and the image of ι is the closure  of A in Ĝ. If A happens to be a direct factor of G, i.e. G = A × B for some normal subgroup B, then the profinite topology on G induces the profinite topology on A, so that the morphism ι is injective and may be used to identify  with Ā; and Ĝ = Ā × . In [D. Goldstein and R. M. Guralnick. The direct product theorem for profinite groups. J. Group Theory9 (2006), 317–322, Question 3.1], Goldstein and Guralnick ask whether the converse holds: if  is a direct factor of Ĝ, does it follow that A is a direct factor of G? (We take the hypothesis to mean: ‘ι is injective and Ā is a direct factor of Ĝ’.)

We show that the answer is ‘no’ in general, but ‘yes’ in a suitably restricted category, namely the virtually polycyclic groups.