Let $F$ be a free group of finite rank. Does $F$ satisfy the ascending chain condition on normal subgroups each of which is a normal closure of one element?

In other words, can there exist elements $r_1,r_2,\dots\in F$ such that $\langle r_i^F \rangle \subsetneqq \langle r_{i+1}^F \rangle$, for all $i \in \mathbb{N}$?

I'm sure the answer should be known, but I have not yet been able to find a reference for it.

Let me add some background.

  • B.H. Neumann [An essay on free products of groups with amalgamations. Philos. Trans. Roy. Soc. London Ser. A 246 (1954), 503–554.] proved that in the free group of rank $3$, $F_3$, there is an infinite strictly ascending sequence of normal subgroups $N_1 \subsetneqq N_2 \subsetneqq \dots$, such that each $N_i$ is the normal closure of two elements in $F_3$. Moreover, there is an automorphism $\alpha \in Aut(F_3)$ such that $N_{i+1}=\alpha(N_i)$, $i \in \mathbb N$.
  • A.M. Brunner [On a class of one-relator groups. Canadian J. Math. 32 (1980), no. 2, 414–420.] constructed a sequence of (pairwise non-isomorphic and Hopfian) one-relator groups $G_0,G_1,\dots$, such that for all $i<j$, $G_j$ is a proper quotient of $G_{i}$ by the normal closure of a single element. (Unfortunately this does not answer my question since the group $G_0$ is not free.)
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    $\begingroup$ In other words, does there exist a "locally 1-relator infinitely presented f.g. group"? $\endgroup$
    – YCor
    Feb 4 '20 at 15:00
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    $\begingroup$ @YCor, I'm not sure "locally $1$-relator" is the right terminology. For any non-Hopfian $1$-relator group $G$ we get a sequence of non-injective group epimorphisms $F \to G_1 \to G_2 \to \dots$, where each $G_i \cong G$, and the limit is an infinitely presented group. But I think that the kernel of the resulting map $F \to G_i$ cannot be the normal closure of a single element, for every $i \in \mathbb{N}$. $\endgroup$ Feb 4 '20 at 15:10
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    $\begingroup$ @YCor "Locally" has an established usage meaning finitely generated subgroups have a given property, so "Locally 1-relator" would be read as f.g. subgroups are 1-relator. $\endgroup$ Feb 4 '20 at 15:20
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    $\begingroup$ @Ashot: note that there are chains of arbitrary finite length: $[x,y, ,y] , ..., x$, I have never seen an infinite chain.. $\endgroup$
    – user6976
    Feb 4 '20 at 18:02
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    $\begingroup$ @Mark and Autumn: here is an even easier example: the relation $[x,y]=1$, of length $4$, follows from the relation $xy^n=1$, of length $n+1$. So arbitrarily long relators can have consequences of length $4$... $\endgroup$ Feb 5 '20 at 13:53

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