Free group in GAP

Question

I know that in the free group $F$ with two generators $x$ and $y$, there is some word $w \in [F,F]$ such that $xy^2=x^{-2} y^{-3} x^{-2}(xy)^5 w$. Is it possible to find $w$ using GAP?

Answer 1

If you just want some expression of $w$ as a product of commutators, then that is not difficult. As I am sure you know, for any element $w$ in a free group $F$, $w \in [F,F]$ if and only if the exponent sum of each of its generators is $0$.

So suppose that $w \in [F,F]$ and let $x$ be its first letter. Then $x^{-1}$ must also occur in $w$, so $w = xux^{-1}v$ for some $u,v \in F$, and hence $w = (xux^{-1}u^{-1})(uv)$ and it suffices to express $uv$ as product of commutators. Since the length of $uv$ is two less than that of $w$, this process terminates. You could certainly do that calculation in GAP. The resulting expression as a product of commutators would have length at most $l(w)/2$.

If you were looking for a word that was the product of the smallest possible number of commutators (or even a close a approximation to that), then that might be a lot harder, and I can only think of very naive brute force approaches to that, which would be impractical for long words.

Answer 2

There is a recent preprint By Fialkovski and Ivanov that describes an algorithm for computing the commutator length of an element in a free group, and it also gives a minimal commutator representation of an element in $F'$.