Geodesic
Commutator width in the first Grigorchuk group
Laurent Bartholdi1 orcid ; Thorsten Groth2 ; Igor Lysenok3
1Universität des Saarlandes, Saarbrücken, Germany
2Georg-August-Universität Göttingen, Germany
3Steklov Mathematical Institute, Moscow, Russia
Groups, geometry, and dynamics, Tome 16 (2022) no. 2, pp. 493-522

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DOI

Let G be the first Grigorchuk group. We show that the commutator width of G is 2: every element g∈[G,G] is a product of two commutators, and also of six conjugates of a. Furthermore, we show that every finitely generated subgroup H≤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.
DOI : 10.4171/ggd/666
Classification : 20-XX, 68-XX
Mots-clés : self-similar groups, quadratic equations, commutator length

Laurent Bartholdi  1   ; Thorsten Groth  2   ; Igor Lysenok  3

1 Universität des Saarlandes, Saarbrücken, Germany
2 Georg-August-Universität Göttingen, Germany
3 Steklov Mathematical Institute, Moscow, Russia 
Laurent Bartholdi; Thorsten Groth; Igor Lysenok. Commutator width in the first Grigorchuk group. Groups, geometry, and dynamics, Tome 16 (2022) no. 2, pp. 493-522. doi: 10.4171/ggd/666
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