Commutator width in the first Grigorchuk group

Let $G$ be the first Grigorchuk group. We show that the commutator width of $G$ is $2$: every element $g\in [G,G]$ is a product of two commutators, and also of six conjugates of $a$. Furthermore, we show that every finitely generated subgroup $H\leq G$ has finite commutator width, which however can be arbitrarily large, and that $G$ contains a subgroup of infinite commutator width. The proofs were assisted by the computer algebra system GAP.

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