On commutator length in free groups

Abstract

Let $F$ be a free group. We present for arbitrary $g\in\mathbb{N}$ a \textsc{LogSpace} (and thus polynomial time) algorithm that determines whether a given $w\in F$ is a product of at most $g$ commutators; and more generally, an algorithm that determines, given $w\in F$, the minimal $g$ such that $w$ may be written as a product of $g$ commutators (and returns $\infty$ if no such $g$ exists). This algorithm also returns words $x\_1,y\_1,\dots,x\_g,y\_g$ such that $w=\[x\_1,y\_1]\dots\[x\_g,y\_g]$. These algorithms are also efficient in practice. Using them, we produce the first example of a word in the free group whose commutator length decreases under taking a square. This disproves in a very strong sense a\~conjecture by Bardakov.