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2022•Groups Geometry and DynamicsOpen access

Commutator width in the first Grigorchuk group

Laurent Bartholdi, Thorsten Groth, Igor Lysenok

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Abstract

Let G be the first Grigorchuk group. We show that the commutator width of G is 2 : every element g\in [G,G] is a product of two commutators, and also of six conjugates of a . Furthermore, we show that every finitely generated subgroup H\leq 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.

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Let G be the first Grigorchuk group. We show that the commutator width of G is 2 : every element g\in [G,G] is a product of two commutators, and also of six conjugates of a . Furthermore, we show that every finitely generated subgroup H\leq 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.

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Available abstract

Let G be the first Grigorchuk group. We show that the commutator width of G is 2 : every element g\in [G,G] is a product of two commutators, and also of six conjugates of a . Furthermore, we show that every finitely generated subgroup H\leq 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.

Key concepts: Mathematics, Commutator, Group (periodic table), Finitely-generated abelian group, Pure mathematics, Algebra over a field, Lie conformal algebra, Organic chemistry

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