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I am working on a problem in Combinatorial Group Theory related to a construction in Algebraic Geometry, and I would like to have a conceptual proof of the fact described below. I am looking for ...

What is the length of the shortest word $w\in F_2$ such that $w(x,y)$ is trivial for every $x,y\in S_n$? There is a simple argument showing that we must have $\ell(w)\geq n$. See here for instance. ...