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Peer-reviewedCC BYGroups, Geometry, and Dynamics

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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Cite

BibTeXRIS
BibTeX
@article{bartholdi2026commutator,
  title = {{Commutator width in the first Grigorchuk group}},
  author = {Laurent Bartholdi and Thorsten Groth and Igor Lysenok},
  journal = {Groups, Geometry, and Dynamics},
  year = {2026},
  month = nov,
  doi = {10.4171/ggd/666},
  url = {https://hal.science/hal-05849726v1},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Commutator width in the first Grigorchuk group
AU  - Laurent Bartholdi
AU  - Thorsten Groth
AU  - Igor Lysenok
PY  - 2026
DA  - 2026/11/03
JO  - Groups, Geometry, and Dynamics
DO  - 10.4171/ggd/666
UR  - https://hal.science/hal-05849726v1
AB  - 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.
SN  - 1661-7207
SN  - 1661-7215
ER  -