2 00 5 the True Prosoluble Completion of a Group Examples and Open Problems

The true prosoluble completion PS(Γ) of a group Γ is the inverse limit of the projective system of soluble quotients of Γ. Our purpose is to describe examples and to point out some natural open problems. We answer the analogue of a question of Grothendieck for profinite completions by providing examples of pairs of non–isomorphic residually soluble groups with isomorphic true prosoluble completions. We also provide a new class of finitely generated examples which answer the original Grothendieck’s problem.