2
$\begingroup$

Let $W=G_p*G_q$, where $G_p$ and $G_q$ are pseudovarieties of all finite $p$-groups and all finite $q$-groups respectively, with $p$ and $q$ fixed prime numbers. The pro-$W$ topology on a group $G$ is the unique group topology such that the set of normal subgroups $N$ with $G/N$ in $W$ is a fundamental system of neighborhoods of the identity.

Let $FG(A)$ be a finitely generated free group. Steinberg and Auinger proved that the pro-$W$ closure of a finitely generated subgroup $H$ of $FG(A)$ is also finitely generated.

Question. Let $H$ be a finitely generated subgroup of a finitely generated free group $FG(A)$. Is there an algorithm to compute a basis for the pro-$W$ closure $H$ given as input a generating set for $H$?

I really stuck in this problem. Of course this is an open problem. Ribes and Zalesskii provided an algorithm for computing a basis for the pro-$p$ closure of a finitely generated subgroup of a free group.

$\endgroup$
1
  • $\begingroup$ This is wide open $\endgroup$ Apr 20, 2015 at 13:34

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.