TAILIEUCHUNG - Đề tài " Grothendieck’s problems concerning profinite completions and representations of groups "

In 1970 Alexander Grothendieck [6] posed the following problem: let Γ1 and Γ2 be finitely presented, residually finite groups, and let u : Γ1 → Γ2 be a ˆ homomorphism such that the induced map of profinite completions u : Γ1 → Γ2 ˆ ˆ is an isomorphism; does it follow that u is an isomorphism? In this paper we settle this problem by exhibiting pairs of groups u : P → Γ such that Γ is a direct product of two residually finite, hyperbolic groups, P is a finitely presented subgroup of infinite index, P is not. | Annals of Mathematics Grothendieck s problems concerning profinite completions and representations of groups By Martin R. Bridson and Fritz J. Grunewald Annals of Mathematics 160 2004 359 373 Grothendieck s problems concerning profinite completions and representations of groups By Martin R. Bridson and Fritz J. Grunewald Abstract In 1970 Alexander Grothendieck 6 posed the following problem let Fl and r2 be finitely presented residually finite groups and let u r1 r2 be a homomorphism such that the induced map of profinite completions u r 1 r2 is an isomorphism does it follow that u is an isomorphism In this paper we settle this problem by exhibiting pairs of groups u P r such that r is a direct product of two residually finite hyperbolic groups P is a finitely presented subgroup of infinite index P is not abstractly isomorphic to r but u P r is an isomorphism. The same construction allows us to settle a second problem of Grothendieck by exhibiting finitely presented residually finite groups P that have infinite index in their Tannaka duality groups c1a P for every commutative ring A 0. 1. Introduction The profinite completion of a group r is the inverse limit of the directed system of finite quotients of T it is denoted by r. If r is residually finite then the natural map r r is injective. In 6 Grothendieck discovered a remarkably close connection between the representation theory of a finitely generated group and its profinite completion if A 0 is a commutative ring and u r1 r2 is a homomorphism of finitely generated groups then u r 1 r2 is an isomorphism if and only if the restriction functor u A RepA r2 RepA r1 is an equivalence of categories where RepA r is the category of finitely presented A-modules with a T-action. Grothendieck investigated under what circumstances u r 1 r2 being an isomorphism implies that u is an isomorphism of the original groups. This led him to pose the celebrated problem Grothendiegk s First Problem. Let r1 and r2 be finitely presented

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