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Let $F_n$ be the free group with $n$ generators $\gamma_1,\dots,\gamma_n$. Consider the homomorphisms $h_i\colon F_n\to F_{n-1}$ defined by adding the relation $\gamma_i=1$ in $F_n$.

What is the least word norm of a nontrivial element $\gamma\in F_n$ such that $h_i(\gamma)=1$ for each $i$?

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    $\begingroup$ One obvious candidate is a nested commutator $[ [\cdots [ [\gamma_1, \gamma_2], \gamma_3], \cdots], \gamma_n ]$. $\endgroup$
    – Sam Hopkins
    Commented Feb 14 at 0:57
  • $\begingroup$ @SamHopkins but is it shortest? $\endgroup$
    – Anton Petrunin
    Commented Feb 14 at 1:02
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    $\begingroup$ math.stackexchange.com/q/2178878/431882 $\endgroup$
    – Denis T
    Commented Feb 14 at 1:26
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    $\begingroup$ $[[\gamma_1,\gamma_2],[\gamma_3,\gamma_4]]$ has length 16, hence shorter than $[[[\gamma_1,\gamma_2],\gamma_3],\gamma_4]$ of length 22. (More generally a quadratic bound follows, much better than the exponential size nested commutator, as observed in the PS-linked paper arxiv.org/abs/1203.3602) $\endgroup$
    – YCor
    Commented Feb 14 at 7:52
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    $\begingroup$ See the link above for a copy of this question, previously answered. The question is not open, but this does not seem to be very well-known. The exact minimum (which is roughly quadratic in $n$) is determined in the paper eudml.org/doc/282667 "Brunnian links" by Gartside and Greenwood. This question also appeared recently again at math.stackexchange.com/questions/4845211/… $\endgroup$
    – Sean Eberhard
    Commented Feb 14 at 9:47

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Repeating from the comments section:

This (natural and beautiful) question was previously asked and answered on this site. See Collapsible group words. It also appeared recently on math.se.

The question is not open, but this does not seem to be well-known. The exact minimum (which is roughly quadratic in $n$), as well as the number and structure of all minimizing elements, was determined in the remarkable paper "Brunnian links" by Gartside and Greenwood. A minimizing element is indeed given by iterated commutators of the generators.

I am not sure why this paper was overlooked by the Demaine et al. survey, given the date that paper appeared and the answers in the linked MO question.

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    $\begingroup$ So, is this oeis.org/A073121 ? $\endgroup$
    – YCor
    Commented Feb 14 at 12:32
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    $\begingroup$ @YCor Yes it is. $\endgroup$
    – Sean Eberhard
    Commented Feb 14 at 14:20
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    $\begingroup$ Precisely, by induction, given $n\ge 2$, write $m=\lceil n/2\rceil$, and let $c_m$, $c_{n-m}$ be minimizing words on $m$ and $n-m\in\{m,m-1\}$ variables. Then $c_n(x_1,\dots,x_n)=[c_m(x_1,\dots,x_m),c_{n-m}(x_{m+1},\dots,x_n)]$ is minimizing, with $c_1(x_1)=x_1$. So $c_2(x,y)=[x,y]$, $c_3(x,y,z)=[[x,y],z]$, $c_4(\dots)=[[x,y],[z,w]]$, $c_5(\dots)=[[[x,y],z],[w,t]]$, etc., yields a minimizing sequence. $\endgroup$
    – YCor
    Commented Feb 14 at 23:51

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