Grothendieck's problem on profinite completions and representations of groups
\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.
finitely generated residually finite groups
scientific article; zbMATH DE number 4190017
scientific article