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PreprintCC BY 4.0arXiv

Small growth rates of free groups

  • Koji Fujiwara
  • Sang-hyun Kim
  • Ryokichi Tanaka
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Abstract

We introduce the notion of a Magnus marking for a finite generating set of a group and prove a certain expansion property. Using this property, we determine the second and third smallest growth rates of the rank-d free group for d≥ 2. We also give a new lower bound for the smallest growth rate of the genus-g surface group for g≥ 2, as well as a lower bound for the growth rate associated with a one-relator presentation.

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Cite

BibTeXRIS
BibTeX
@misc{fujiwara2026small,
  title = {{Small growth rates of free groups}},
  author = {Koji Fujiwara and Sang-hyun Kim and Ryokichi Tanaka},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.17444},
  eprint = {2607.17444},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.17444},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Small growth rates of free groups
AU  - Koji Fujiwara
AU  - Sang-hyun Kim
AU  - Ryokichi Tanaka
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2607.17444
UR  - https://arxiv.org/abs/2607.17444
AB  - We introduce the notion of a Magnus marking for a finite generating set of a group and prove a certain expansion property. Using this property, we determine the second and third smallest growth rates of the rank-d free group for d≥ 2. We also give a new lower bound for the smallest growth rate of the genus-g surface group for g≥ 2, as well as a lower bound for the growth rate associated with a one-relator presentation.
ER  -