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PreprintarXiv licencearXiv

Asymptotic dimension of commutative monoid actions

  • Jinmin Wang
  • Jing Yu
  • Jingming Zhu
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Abstract

We prove that every bounded-to-one action of a finitely generated commutative monoid has asymptotic dimension at most the torsion-free rank of its group completion, with control uniform for a fixed monoid, generating set, and generator fiber bound. The estimate is sharp. Together with the equality of classical and Borel asymptotic dimensions for these actions, it gives the sharp Borel rank bound and recovers hyperfiniteness for bounded-to-one Borel actions of countable commutative monoids.

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Cite

BibTeXRIS
BibTeX
@misc{wang2026asymptotic,
  title = {{Asymptotic dimension of commutative monoid actions}},
  author = {Jinmin Wang and Jing Yu and Jingming Zhu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26954},
  eprint = {2609.26954},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26954},
}
RIS
TY  - UNPB
TI  - Asymptotic dimension of commutative monoid actions
AU  - Jinmin Wang
AU  - Jing Yu
AU  - Jingming Zhu
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.26954
UR  - https://arxiv.org/abs/2609.26954
AB  - We prove that every bounded-to-one action of a finitely generated commutative monoid has asymptotic dimension at most the torsion-free rank of its group completion, with control uniform for a fixed monoid, generating set, and generator fiber bound. The estimate is sharp. Together with the equality of classical and Borel asymptotic dimensions for these actions, it gives the sharp Borel rank bound and recovers hyperfiniteness for bounded-to-one Borel actions of countable commutative monoids.
ER  -