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Grothendieck’s problems concerning profinite completions and representations of groups. (English) Zbl 1083.20023

This relevant paper deals with some questions posed by A. Grothendieck [Manuscr. Math. 2, 375-396 (1970; Zbl 0239.20065)] in his study of the connection between the representation theory of a finitely generated group and its profinite completion. The celebrated Grothendieck’s first problem is the following: let \(u\colon\Gamma_1\to\Gamma_2\) be a homomorphism of finitely presented, residually finite groups such that its extension to their profinite completions \(\widehat u\colon\widehat\Gamma_1\to\widehat\Gamma_2\) is an isomorphism; must \(u\) be an isomorphism as well?
A negative solution to the corresponding problem for finitely generated groups was given by Platonov and Tavgen; other counterexamples were discovered by Bass, Lubotzky and Pyber but in all these cases the groups involved were not finitely presented, as requested by Grothendieck.
In this paper the authors settle the question in the negative. They exhibit a residually finite, 2-dimensional, hyperbolic group \(H\) and a finitely presented subgroup \(P\) of \(\Gamma=H \times H\) of infinite index, such that \(P\) is not abstractly isomorphic to \(\Gamma\), but the inclusion \(u\colon P\to\Gamma\) induces an isomorphism \(\widehat u\colon\widehat P\to\widehat\Gamma\).
The same construction allows the authors to settle a second problem of Grothendieck: let \(\Gamma\) be a finitely presented, residually finite group; is the natural monomorphism from \(\Gamma\) to the Tannaka duality group \(\text{cl}_A(\Gamma)\) an isomorphism for every nonzero commutative ring \(A\), or at least some suitable commutative ring \(A\neq 0\)? Also in this case the answer is negative; indeed, if \(P\) is as above, then \(P\) is of infinite index in \(\text{{cl}}_A(P)\) for every commutative ring \(A \neq 0\).

MSC:

20E18 Limits, profinite groups
20E26 Residual properties and generalizations; residually finite groups
20F05 Generators, relations, and presentations of groups

Citations:

Zbl 0239.20065
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