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> What is the length $f(n)$ of the shortest nontrivial group word $w\_n$ in $x\_1,\ldots,x\_n$ that collapses to $1$ when we substitute $x\_i=1$ for any $i$?
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For example, $f(2)=4$, with the commutator $[x\_1,x\_2]=x\_1 x\_2 x\_1^{-1} x\_2^{-1}$ attaining the bound. 


For any $m,n \ge 1$, the construction $w\_{m+n}(\vec{x},\vec{y}):=[w\_m(\vec{x}),w\_n(\vec{y})]$ shows that $f(m+n) \le 2 f(m) + 2 f(n)$.


Is $f(1),f(2),\ldots$ the same as sequence [A073121](http://oeis.org/A073121 "A073121"):
$$ 1,4,10,16,28,40,52,64,88,112,136,\ldots ?$$


**Motivation:** Beating the iterated commutator construction would improve the best known bounds in [size of the smallest group not satisfying an identity](https://mathoverflow.net/questions/15022/size-of-the-smallest-group-not-satisfying-an-identity/15065#15065).



Collapsible group words
See the paper "Brunnian links" by Gartside and Greenwood, published in Fundamenta Mathematicae. Theorems 8 and 7 imply that iterated commutators are optimal and the sequence you suggest gives the minimal length. 


