Reviewing old qualifying exams in algebraic topology, I see a lot of the following type of question: let $F$ be the free group on generators $a$ and $b$. Describe a set of free generators of the subgroup generated by all conjugates of $S$, where $S$ is some small set of elements in $F$.
EDIT: For a concrete example, I've seen a solution when $S = \{aba^{-1}b^{-1}\}$. I have seen the Cayley graph that is drawn, but am not sure why they chose that specific graph.
I know that I have to draw a certain Cayley (Schreier?) graph, but as it is phrased I don't know which one to draw. For instance, some questions like this are phrased slightly differently: let $h : F \to G$ be a surjective homomorphism sending $a \mapsto x, b \mapsto y$ where $x, y$ are generators of $G$. Then I know that I should be drawing the Cayley graph of the group $G$. However, when the question is phrased like in the above paragraph, I am at a loss. Perhaps to be more clear, the involvement of the phrase "all conjugates of" throws me off.
Also, is there any way to know a priori if the given subgroup is normal? Or would I have to check that the relevant covering is normal/regular?
Best Answer
I am afraid that this answer might not be what you are looking for, because I am thinking algebraically rather than topologically.
In your example, the subgroup is the commutator subgroup $[F,F]$. The vertices of the Schreier graph can be labelled by a transversal to the subgroup, and the obvious choice is $\{a^ib^j: i,j \in {\mathbb Z}\}$. You have edges labelled $a$ and $b$ from $a^ib^j$ to $a^{i+1}b^j$ and $a^ib^{j+1}$, respectively.
The associated free subgroup generators (often called the Schreier generators) are $\{ a^ib^jab^{-j}a^{-i-1} : i,j \in {\mathbb Z}, j \ne 0 \}$.
In a general problem of this type, the vertices of the Schreier graph correspond to the elements of the quotient group $F/\langle S^F \rangle$ of $F$, which is the group defined by the presentation $\langle a,b \mid S \rangle$.