Profinite Completions and Continuous Extensions of Morphisms Between Free Groups and Theirs Profinite Completions

It is known that every morphism φ : F → F' between free groups of pseudovariety V of finite groups is uniformly continuous when both groups are equipped with their respective pro-V topologies. In this paper we prove that this morphism can be uniquely extended to a continuous morphism between their pro-V completions φ̂ : F̂ → F'̂? 2000 Mathematics Subject Classification. 20E18; 20E05