Intersection of descending series in a free group

Question

I have stumbled upon a problem. It can be stated in the following way: Let $E$ be a finitely generated free group. Denote $\gamma_n(E)$ the $n$-th term of the lower central series. Consider a generator in this group $e\in E$. I am interested in finding subsets of the set $$\cap_{i=1}^\infty e\gamma_n(E).$$

I have tried the following approach: the set $e\gamma_n(E)$ is obviously a subset of $\langle e, \gamma_n(E)\rangle$. In the question Intersection of subgroup of a free group with the lower central series I have seen an answer stated this way: the sequence $\langle S,\gamma_n(E)\rangle$ converges to $S$ iff $S$ is closed in the pronilpotent topology on the group $E$. However, it is not really clear whether $\{e\}$ is closed and, if not, how to compute the said intersection.

Singletons are closed on the pronilpotent topology because free groups are residually finite p-groups — Benjamin Steinberg, Apr 03 '21 at 16:11
Thank you! Maybe you have in mind an article of some sort where I could read about this? — John Conway, Apr 03 '21 at 16:22
You can find that in lots of places. The easiest proof is via the Magnus representation. I think it is in Ribes and Zalesskii and probably it is in math overflow — Benjamin Steinberg, Apr 03 '21 at 16:28
See https://mathoverflow.net/questions/20471/why-are-free-groups-residually-finite — Benjamin Steinberg, Apr 03 '21 at 18:16
If John Conway is not your real name you might be asked to change the username (if your name is John Conway, you might consider adding a middle letter). See this Meta question — YCor, Apr 03 '21 at 18:46
I don't know what you mean by "generator" in a given group. Maybe you mean some element of a fixed free generating subset? — YCor, Apr 03 '21 at 18:47
I have downloaded the book by Ribes and Zalesskii, thank you. Concerning generators: yes, I meant an element of a basis of $E$. — John Conway, Apr 03 '21 at 18:52
An(other) explicit reference for free groups being residually (finite $p$-groups) is Theorem 5.2 of Hall, Marshall A topology for free groups and related groups, Ann. Math. (2) 52, 127-139 (1950). ZBL0041.36210. — Carl-Fredrik Nyberg Brodda, Apr 03 '21 at 21:06

Intersection of descending series in a free group

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