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...)
- 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).