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PreprintarXiv licencearXiv

Irrationality of the conjugacy growth on Heisenberg groups

  • Lukas Vandeputte
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Abstract

We demonstrate that the conjugacy growth series for the Heisenberg groups Hn(ℤ) is never algebraic, irrespective of the generating set. This makes progress towards the conjecture that all finitely presented groups that are not virtually abelian have irrational conjugacy growth series. Contrary to the group H3(ℤ), where the conjugacy growth is n2log(n), the conjugacy growth of the other Heisenberg groups is polynomial, making these groups the first known nilpotent groups where the irrationality does not follow from the asymptotic growth.

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Cite

BibTeXRIS
BibTeX
@misc{vandeputte2026irrationality,
  title = {{Irrationality of the conjugacy growth on Heisenberg groups}},
  author = {Lukas Vandeputte},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.37334},
  eprint = {2610.37334},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.37334},
}
RIS
TY  - UNPB
TI  - Irrationality of the conjugacy growth on Heisenberg groups
AU  - Lukas Vandeputte
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.37334
UR  - https://arxiv.org/abs/2610.37334
AB  - We demonstrate that the conjugacy growth series for the Heisenberg groups H_n(ℤ) is never algebraic, irrespective of the generating set. This makes progress towards the conjecture that all finitely presented groups that are not virtually abelian have irrational conjugacy growth series. Contrary to the group H_3(ℤ), where the conjugacy growth is n^2log(n), the conjugacy growth of the other Heisenberg groups is polynomial, making these groups the first known nilpotent groups where the irrationality does not follow from the asymptotic growth.
ER  -