Commutator width in the first Grigorchuk group
Laurent Bartholdi
Universität des Saarlandes, Saarbrücken, GermanyThorsten Groth
Georg-August-Universität Göttingen, GermanyIgor Lysenok
Steklov Mathematical Institute, Moscow, Russia
Abstract
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.
Supplementary Material
Cite this article
Laurent Bartholdi, Thorsten Groth, Igor Lysenok, Commutator width in the first Grigorchuk group. Groups Geom. Dyn. 16 (2022), no. 2, pp. 493–522
DOI 10.4171/GGD/666