Commutator width in the first Grigorchuk group

TL;DR

This paper investigates the commutator width of the first Grigorchuk group, establishing it as 2, and explores properties of finitely generated subgroups and subgroups with infinite commutator width, using computational assistance.

Contribution

It proves the exact commutator width of the first Grigorchuk group and analyzes the commutator width of its subgroups, including the existence of subgroups with infinite width.

Findings

Commutator width of G is exactly 2.

Finitely generated subgroups have finite but arbitrarily large commutator width.

G contains a subgroup with infinite commutator width.

Abstract

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.