Let $m,n \in \mathbb{N}$. Which residually finite groups $G$ generated by $m$ elements, have the free profinite group on $n$ generators as their profinite completion?
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$\begingroup$ I believe this is a more general version of a well known open question. See page 4 of arxiv.org/pdf/1108.5130.pdf $\endgroup$– Khalid Bou-RabeeCommented Apr 25, 2014 at 11:49
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$\begingroup$ I'm aware of this paper. I do not think that this problem receives satisfactory treatment there. $\endgroup$– PabloCommented Apr 25, 2014 at 12:44
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$\begingroup$ Khalid is right, though, that this is a well known open question. I recently heard Alan Reid give a talk about his joint work with Martin Bridson on this subject ma.utexas.edu/users/areid/BCR_website.pdf. Thm 1.2 gives examples of classes of groups whose profinite completions are not isomorphic to free profinite groups. $\endgroup$– Benjamin SteinbergCommented Apr 25, 2014 at 15:33
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