If I have a subgroup $S$ of a free group $\mathcal{F}_m$, what can I say about the behaviour of the descending sequence of subgroups $\left< S, \Gamma_c(\mathcal{F}_m) \right>$ (where $\Gamma_c(\mathcal{F}_m)$ is the $c$-th term of the lower central series)?
In particular, does this sequence always converge to $S$ or not?
(Below is what I've tried so far)
If after some point $S$ contains $\Gamma_c$, then this is obviously true, but that's only the case if the corresponding f.g. group $\mathcal{F}_m/S$ is nilpotent, so there's still a lot to go. If after some point $\Gamma_c \cap S =\{1\}$ then it's also true, but that would mean $S$ is nilpotent which is impossible since $S$ is free as a subgroup of a free group.
I've tried to use the diamond isomorphism theorem to use that $S \Gamma_c/S \cong \Gamma_c/S \cap \Gamma_c$, but it doesn't seem clear to me that my sequence converging to $S$ has much of a connection with $S \Gamma_c/S$ converging to the trivial group - indeed it's not entirely clear what "converging" means in the context of these quotients.
Thanks a lot!