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PreprintCC BY-NC-ND 4.0arXiv

Are cluster automorphism groups finitely generated?

  • Changjian Fu
  • Zhanhong Liang
  • Yinzhi Wang
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Abstract

This paper investigates the finite generation of cluster automorphism groups. By applying the pseudo ℕ-grading introduced in our previous work, we establish a sufficient condition for a cluster automorphism group to be finitely generated. As applications, we re-establish the finite generation of the automorphism groups for all finite mutation type cluster algebras, and verify the acyclic cases. Furthermore, we illustrate through examples that our approach significantly simplifies the computation of presentations for these groups in certain cases.

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Cite

BibTeXRIS
BibTeX
@misc{fu2026cluster,
  title = {{Are cluster automorphism groups finitely generated?}},
  author = {Changjian Fu and Zhanhong Liang and Yinzhi Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.16854},
  eprint = {2605.16854},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.16854},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Are cluster automorphism groups finitely generated?
AU  - Changjian Fu
AU  - Zhanhong Liang
AU  - Yinzhi Wang
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2605.16854
UR  - https://arxiv.org/abs/2605.16854
AB  - This paper investigates the finite generation of cluster automorphism groups. By applying the pseudo ℕ-grading introduced in our previous work, we establish a sufficient condition for a cluster automorphism group to be finitely generated. As applications, we re-establish the finite generation of the automorphism groups for all finite mutation type cluster algebras, and verify the acyclic cases. Furthermore, we illustrate through examples that our approach significantly simplifies the computation of presentations for these groups in certain cases.
ER  -