Direct Factors of Profinite Completions and Decidability

We consider finitely presented, residually finite groups G and finitely generated normal subgroups A such that the inclusion A ↪→ G induces an isomorphism from the profinite completion of A to a direct factor of the profinite completion of G. We explain why A need not be a direct factor of a subgroup of finite index in G; indeed G need not have a subgroup of finite index that splits as a non-trivial direct product. We prove that there is no algorithm that can determine whether A is a direct factor of a subgroup of finite index in G. Let G be a finitely generated residually finite group. The inclusion A ↪→ G of any finitely generated subgroup induces a morphism of profinite completions ι : Â → Ĝ. If A is a direct factor of G then ι is injective and we can identify the closure A of ι(A) with Â. In [13] Nikolov and Segal answered a question of Goldstein and Guralnick [11] by showing that the converse of the preceding observation is false: there exist pairs of finitely generated residually finite groups A ↪→ G, with A is normal in G, such that ι : Â→ A is an isomorphism, A is a direct factor of Ĝ, but A is not a direct factor of G, nor indeed of any subgroup of finite index in G. Nikolov and Segal proved this by exhibiting an explicit group of the form G = Aoα Z, where A is finitely generated and α, although not inner, induces an inner automorphism on A/N for every α-invariant subgroup of finite index N ⊂ A. The first purpose of the present note is to explain how pairs of residually finite groups A ↪→ G settling the Goldstein-Guralnick question also arise from the constructions in [7]. As well as providing a broader range of examples, these constructions allow one to impose extra conditions on A and G (see subsection 1.2). For example, one can require G to be finitely presented, indeed to be a direct product of torsion-free hyperbolic groups and hence have a finite classifying space. If one drops the requirement that A be normal, one can arrange for both A and G to be finitely presented. Date: 19 March 2008. 2000 Mathematics Subject Classification. 20E18, 20F10.