Grothendieck's problem on profinite completions and representations of groups
type: Item
instance of: scholarly article
title: Grothendieck's problem on profinite completions and representations of groups
zbMATH Open document ID: 0722.20020
DOI: 10.1007/BF00534194
author: O. I. Tavgen'
author: V. P. Platonov
published in: \(K\)-Theory
publication date: 1990-01-01T00:00:00Z
review text: \textit{A. Grothendieck} [Manuscr. Math. 2, 375-396 (1970; Zbl 0239.20065)] posed a problem on profinite completions which is presented in this paper as Problem 1: Let u: \(G_ 1\to G_ 2\) be a homomorphism of finitely generated residually finite groups such that its extension to their profinite completions \(\hat u:\) \(\hat G_ 1\to \hat G_ 2\) is an isomorphism. Must u be an isomorphism as well? The authors of this paper found a counterexample [Dokl. Akad. Nauk SSSR 288, 1054-1058 (1986; Zbl 0614.20016)], so Grothendieck's problem is solved negatively in the general case. But in this example G is not finitely presented, hence Problem 1 still remains open for finitely presented groups. In the second part of the paper, the connection of Grothendieck's problem with representation theory are discussed. Platonov has raised the following two problems. Problem 2: Let \(G_ 2\) be an arithmetic (S- arithmetic) subgroup of an algebraic group G defined over a number field k. Suppose that the congruence kernel \(C^ 2(G(K))\) is finite. Then does Problem 1 have an affirmative solution for \(G_ 2?\) Problem 3: Let \(\Gamma\) be a finitely generated linear group such that dim \(X_ n(\Gamma)=0\) for n where \(X_ n(\Gamma)\) is the algebraic variety of all n-dimensional characters. Is \(\Gamma\) isomorphic to a group of arithmetic type, i.e. commensurable with a direct product of a finite number of S- arithmetic groups? In this part, it is proved that when \(G_ i\) are linear groups and dim \(X_ n(G_ i)=0\) for any n, then the positive solution of Problem 3 implies the positive solution of Problem 1 for u. It is also shown that if \(G\subset GL_ m(Z_ S)\) such that dim \(X_ n(G)=0\) for any n, and G and its Zariski closure \(\bar G(Z_ S)\) have finite congruence kernels then if Problem 2 has a positive solution, then G is arithmetic.
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 20G30
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20E26
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 4190017
zbMATH Keywords: profinite completions
zbMATH Keywords: finitely generated residually finite groups
zbMATH Keywords: Grothendieck's problem
zbMATH Keywords: algebraic group
zbMATH Keywords: number field
zbMATH Keywords: algebraic variety
zbMATH Keywords: n-dimensional characters
zbMATH Keywords: group of arithmetic type
zbMATH Keywords: S-arithmetic groups
zbMATH Keywords: congruence kernels
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00534194
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description: scientific article; zbMATH DE number 4190017