Let be the first Grigorchuk group. We show that the commutator width of is : every element is a product of two commutators, and also of six conjugates of . Furthermore, we show that every finitely generated subgroup has finite commutator width, which however can be arbitrarily large, and that contains a subgroup of infinite commutator width. The proofs were assisted by the computer algebra system GAP.
Cite this article
Laurent Bartholdi, Thorsten Groth, Igor Lysenok, Commutator width in the first Grigorchuk group. Groups Geom. Dyn. (2022), published online firstDOI 10.4171/GGD/666