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This is an (early) exercise from the book "Analytic Pro-p groups": (p.31, ex. 3(iii))

Give an example of a finitely generated pro-$p$ group $G$ and a dense subgroup $H$ of $G$, with $H$ finitely generated as an abstract group , such that $\hat{H_p} \ncong G$.

Any ideas would be really appreciated. I do suspect that the answer would be a linear group though.

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  • $\begingroup$ This is exercise 3 (iii), p. 31 in Dixon, du Sautoy, Mann, Segal: Analytic Pro-$p$ Groups. The condition is $\hat H_p \not\cong G$. $\endgroup$
    – BIS HD
    Commented Nov 20, 2013 at 11:08

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Just to kill a mosquito with a cannonball: a theorem of Abert states that every profinite weakly branch group contains a dense free subgroup. For example, the pro-2 completion of Grigorchuck's group is (weakly) branch but it is not free (e.g. by subgroup growth considerations).

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