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If $F$ is a free discrete group, then any subgroup $H$ of $F$ is free: this is the well-known theorem of Nielsen-Schreier. Moreover, there is a well-known algorithm, the Nielsen-Schreier method that allows us to determine a set of free generators of $H$ (see e.g. this reference).

Now if $F$ is free pro-$p$-group, then any closed subgroup $H$ of $F$ is pro-$p$-free (cf Ribes and Zaleskii, Profinite groups, Cor. 7.7.5). But is there a method to describe a space on which $H$ is free?

I am interested in the question even in the perhaps much simpler case where $F$ is the pro-p-completion of a free discrete group $F_0$ (with finitely many generators), and $H$ is the closure in $F$ of a subgroup $H_0$ of $F_0$ (of which I happen to know a set of free generators --infinitely many-- explicitly by the Nielsen-Schreier methods). Then by right-exacness of the $p$-completion functor, $H$ is a quotient of the completion of $H_0$, but I can't see by what (if anything).

This question is a follow-up of this one.

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This doesn't quite answer your question, but... If $F$ is a discrete free group and $H$ is a fg subgroup, Ribes and Zalesskii gave an algorithm, improved by Margolis, Sapir and Weil, to compute the pro-p closure K of H in F. In this case K is fg and its closure in the pro-p completion of F is its pro-p completion. I am not sure when H is not fg what happens.

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So, if I understand you well, if $H \hookrightarrow F$ is an injective morphism of discrete free groups, with $H$ finitely generated, then the map $\hat{H} \rightarrow \hat{F}$ is also injective, where $\hat{X}$ is the pro-$p$-completion of a group $X$. Is that right? – Joël Dec 9 '11 at 3:25
If the image of H is closed and we are in the pro-C setting with C an extension-closed variety. Closed subgroups inherit their full pro-C topology in this case. – Benjamin Steinberg Dec 9 '11 at 4:23

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