The *pro-solvable topology* on a group $G$ is the unique group topology such that the set of normal subgroups $N\lhd G$ with $G/N$ a finite solvable group is a fundamental system of neighborhoods of the identity.

Let $F$ be a finitely generated free group. Ribes and Zalesskii proved that the pro-solvable closure $\overline H$ of finitely generated subgroup $H$ is also finitely generated (and in fact, the rank of $\overline H$ is at most the rank of $H$).

Question.Let $H$ be a finitely generated subgroup of a finitely generated free group $F$. Is there an algorithm to compute a basis (or equivalently a generating set) for the pro-solvable closure $\overline H$ given as input a generating set for $H$?

I have a number of remarks. First of all this is a well-known *open* problem to people working in finite semigroup theory and the algebraic theory of automata and it has a number of applications to these fields. However, I am not sure that this question is very well known to group theorists, who would be best equipped to solve it.

**What is known.**

This problem is Turing equivalent to each of the following problems.

- Deciding if a finitely generated subgroup of a finitely generated free group is dense in the pro-solvable topology.
- Deciding if a finitely generated subgroup of a finitely generated free group is closed in the pro-solvable topology.
- Deciding if a graph immersion over a wedge of circles can be extended to a finite-sheeted covering such that the permutation group obtained by making the monodromy action faithful is solvable.
- Given a finite monoid $M$, deciding if there exists a finite solvable group $G$ and a submonoid $N$ of the power monoid $P(G)$ such that $M$ is a quotient of $N$. (OK, I don't expect anybody to solve this problem using this characterization, but...)

**Related results.**

- If $p$ is prime, then Ribes and Zalesskii provided an algorithm for computing a basis for the pro-$p$ closure of a finitely generated subgroup of a free group.
- Margolis, Sapir and Weil showed the pro-$p$ closure can be computed in polynomial time (I think a quintic bound was given).
- Margolis, Sapir and Weil showed that a basis for the pro-nilpotent closure of a finitely generated subgroup of a free group can be computed and also solved 1--3 for the nilpotent case (they are not
*a priori*equivalent because nilpotent groups are not closed under extension).