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Let $G_p$ be pseudovariety of all finite $p$-group. The pro-$p$ topology on a group $G$ is the unique group topology such that the set of normal subgroups $N$ with $G/N$ in $G_p$ is a fundamental system of neighborhoods of the identity.

Margolis, Sapir and Weil showed that if a finitely generated subgroup $H$ of the free group is $G_p$-dense then $H$ contains a subgroup of rank $n$(=rank of free group) which is $G_p$-dense. In the proof of theorem 6.1, in "Dynamics of implicit operations and tameness of pseudovarieties ", Jorge Almeida used this fact in every extension closed pseudovarieties. I can not proof this and I think this not true. I appreciate any proof or counter example

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  • $\begingroup$ I expect "$G_p$-dense" means "dense in the pro-$p$ topology? and that the generalization means the same fact when the pro-$p$ topology is replaced with the pro-$V$ topology when $V$ is an extension-closed pseudo variety of finite groups? $\endgroup$
    – YCor
    Commented Jun 19, 2015 at 18:05
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    $\begingroup$ @YCor, this is surely what the op meant $\endgroup$
    – Benjamin Steinberg
    Commented Jun 19, 2015 at 18:25

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