Commutator width in the first Grigorchuk group

  • Laurent Bartholdi

    Universität des Saarlandes, Saarbrücken, Germany
  • Thorsten Groth

    Georg-August-Universität Göttingen, Germany
  • Igor Lysenok

    Steklov Mathematical Institute, Moscow, Russia
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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.

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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