Let l be a prime number. In this present paper, we study the outer Galois action on the profinite and the relative pro-l completions of mapping class groups of pointed orientable topological surfaces. In the profinite case, we prove that the outer Galois action is faithful. In the pro-l case, we prove that the kernel of the outer Galois action has certain stability properties with respect to th...

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 complet...

We prove that if a Cartesian product of alternating groups is topologically finitely generated, then it is the profinite completion of a finitely generated residually finite group. The same holds for Cartesian producs of other simple groups under some natural restrictions.

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 complet...

We show that the profinite completions and canonical extensions of bounded distributive lattices and of Boolean algebras coincide. We characterize dual spaces of canonical extensions of bounded distributive lattices and of Heyting algebras in terms of Nachbin order-compactifications. We give the dual description of the profinite completion ̂ H of a Heyting algebra H, and characterize the dual sp...