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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 $i$?

Anton Petrunin
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    One obvious candidate is a nested commutator $[ [\cdots [ [\gamma_1, \gamma_2], \gamma_3], \cdots], \gamma_n ]$. – Sam Hopkins Feb 14 '24 at 00:57
  • @SamHopkins but is it shortest? – Anton Petrunin Feb 14 '24 at 01:02
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    https://math.stackexchange.com/q/2178878/431882 – Denis T Feb 14 '24 at 01:26
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    $[[\gamma_1,\gamma_2],[\gamma_3,\gamma_4]]$ 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 https://arxiv.org/abs/1203.3602) – YCor Feb 14 '24 at 07:52
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    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 https://eudml.org/doc/282667 "Brunnian links" by Gartside and Greenwood. This question also appeared recently again at https://math.stackexchange.com/questions/4845211/morphism-on-the-commutator-subgroup-of-the-free-group-picture-hanging-puzzle – Sean Eberhard Feb 14 '24 at 09:47

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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.

Sean Eberhard
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    So, is this https://oeis.org/A073121 ? – YCor Feb 14 '24 at 12:32
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    @YCor Yes it is. – Sean Eberhard Feb 14 '24 at 14:20
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    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. – YCor Feb 14 '24 at 23:51