JournalsggdOnline First3 August 2022

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
Commutator width in the first Grigorchuk group cover
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Abstract

Let GG be the first Grigorchuk group. We show that the commutator width of GG is 22: every element g[G,G]g\in [G,G] is a product of two commutators, and also of six conjugates of aa. Furthermore, we show that every finitely generated subgroup HGH\leq G has finite commutator width, which however can be arbitrarily large, and that GG 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. (2022), published online first

DOI 10.4171/GGD/666