A question on Hawaiian earring group I have asked this question in MSE but have not got any satisfactory answer, so I am asking it here. Any idea on how to approach this problem will be highly appreciated.
Consider the Hawaiian earring. Suppose $f_n$'s are the loops representing the circles of radius $1/n$ centered at $(1/n, 0)$ for $n=1,2\cdots$. Suppose the following holds in the fundamental group
$$\langle [f_1, f_2][f_3, f_4]\cdots\rangle = \langle [g_1, g_2]\cdots [g_{2k-1},g_{2k}]\rangle $$
for some loops $g_1,g_2,\ldots , g_{2k}$,
where $\langle\cdot\rangle$ denotes the homotopy class and $[a,b]=aba^{-1}b^{-1}$ denotes the commutator. Then can we say that 
$$\langle [f_1, f_2]\cdots [f_{2n-1}, f_{2n}]\rangle = \langle [g_1, g_2]\cdots [g_{2k-1},g_{2k}]\rangle$$
for all sufficiently large $n$, or for some $n>k$ ?
I had this question while I was reading the paper here (see page 76, last paragraph), where the above has been mentioned (in some other equivalent form) without proof.
 A: This question seems to assume that the infinite product loop $[f_1,f_2][f_3,f_4][f_5,f_6]\dots$ is null-homologous in the Hawaiian earring, which is false. In fact, it is false by Katusya Eda's $0$-form Lemma, which says that a loop $\alpha$ in the Hawaiian earring is null-homologous if and only if the reduced representative of $\alpha$ factors into a finite concatenation $\prod_{i=1}^{2n}\alpha_i$ where there is an inverse pairing among the factors. Formally, this means $\{1,2,\dots ,2n\}$ splits into the disjoint union of two $n$-element sets $A,B$ and there is a bijection $\phi:A\to B$ such that for each $i\in A$, $\alpha_i$ is a reparameterization of the reverse of $\alpha_{\phi(i)}$.
The $0$-form Lemma can be found as Lemma 3.6 in Singular homology groups of one-dimensional Peano continua. Another nice generalization for the Hawaiian earring is Lemma 4.3 in Cotorsion-free groups from a topological point of view by Eda and Fischer. The double induction of this proof is rather subtle and is a thing of beauty.
A: I think your question is interesting and deserves to be solved. But, I cannot find the corresponding part in Higman's paper. For  $c$  in  $G'$, let $r(c)$ be the minimal number $n$  such that  $c = [u_1,v_1]\dots[u_n,v_n]$. Let  $e_i$  be the generator of the $i$-th factor of  $F$. Then, $r([e_1,e_2]...[e_{2n-1},e_{2n}]) = n$, which is known but is not a straightforward fact. Therefore, $[e_1,e_2]\dots[e_{2n-1},e_{2n}]\dots$ does not belong to the commutator subgroup.  
A: I believe this sort of question is addressed in the beautiful series of papers by Cannon and Conner (e.g., the first one:
MR1775709 (2001g:20020) Reviewed 
Cannon, J. W.(1-BYU); Conner, G. R.(1-BYU)
The combinatorial structure of the Hawaiian earring group. (English summary) 
Topology Appl. 106 (2000), no. 3, 225–271. 
I am pretty sure your question or some variant is covered in Section 4 of this.
