On commutator length in free groups

TL;DR

This paper introduces a polynomial-time algorithm to determine the minimal number of commutators needed to express a word in a free group, and provides the first example of a word whose commutator length decreases when squared, disproving a conjecture.

Contribution

It presents the first efficient algorithm for computing commutator length in free groups and provides a counterexample to a longstanding conjecture.

Findings

Algorithm runs in LogSpace and polynomial time.

First example of a word with decreasing commutator length when squared.

Disproves Bardakov's conjecture.

Abstract

Let $F$ be a free group. We present for arbitrary $g\in\mathbb{N}$ a 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). The algorithm also returns words $x_1,y_1,\dots,x_g,y_g$ such that $w=[x_1,y_1]\cdots[x_g,y_g]$. The algorithms we present 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.