Grothendieck's problems concerning profinite completions and representations of groups

• 成果类型: Article
• 署名作者: Bridson, MR; Grunewald, FJ
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2004.160.359
• 发表日期: 2004
• 页码: 359-373
• 关键词: discrete-groups

摘要

In 1970 Alexander Grothendieck [6] posed the following problem: let Gamma(1) and Gamma(2) be finitely presented, residually finite groups, and let u : Gamma(1) --> Gamma(2) be a homomorphism such that the induced map of profinite completions u : Gamma(1) --> Gamma(2) 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 --> Gamma such that Gamma 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 Gamma, but u : P --> Gamma 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 cl(A)(P) for every commutative ring A not equal 0.