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Let $V$ be a extension closed pseudovariety. The pro-$V$ topology on a group $G$ is the unique group topology such that the set of normal subgroups $N$ with $G/N$ in $V$ is a fundamental system of neighborhoods of the identity. Let $K$ be a finitely generated subgroup of finite index of free group $FG(A)$ which is dense respect to pro-$V$ topology on the free group. Is it true that $K$ contains a subgroup of rank n(=rank of free group) which is $V$-dense?

user182085
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  • Unless I misunderstood your question, this fails trvially for the profinite topology, since any subgroup of finite index is closed. – HJRW Jul 24 '15 at 11:16
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    Sorry I forgot to write $K$ is finitely generated subgroup. So in profinite topology this is true. – user182085 Jul 24 '15 at 11:19
  • Ah right, I misunderstood you. Yes, it is true in the pro finite topology. – HJRW Jul 24 '15 at 15:53
  • Also this true in pro p topology when p is a prime number. But I do not know for other extension closed pseudovariety – user182085 Jul 24 '15 at 17:13

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