TITLE: Collapsible group words
QUESTION [45 upvotes]: 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 $i$?

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:
$$ 1,4,10,16,28,40,52,64,88,112,136,\ldots ?$$
Motivation: Beating the iterated commutator construction would improve the best known bounds in size of the smallest group not satisfying an identity.

REPLY [19 votes]: 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.