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Taming Reidemeister zeta functions on virtually polycyclic groups

  • Sam Tertooy
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Abstract

We prove a formula for Reidemeister numbers of virtually polycyclic groups that generalises known formulas for finite groups and torsion-free virtually polycyclic groups. Based on this formula, we introduce a new invariant which we call the tamed Reidemeister number, and prove that its associated zeta functions are rational on virtually polycyclic groups. We also show that this implies rationality of (ordinary) Reidemeister zeta functions, as well as rationality of Nielsen zeta functions on compact infra-solvmanifolds.

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Cite

BibTeXRIS
BibTeX
@misc{tertooy2026taming,
  title = {{Taming Reidemeister zeta functions on virtually polycyclic groups}},
  author = {Sam Tertooy},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.24935},
  eprint = {2609.24935},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24935},
}
RIS
TY  - UNPB
TI  - Taming Reidemeister zeta functions on virtually polycyclic groups
AU  - Sam Tertooy
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.24935
UR  - https://arxiv.org/abs/2609.24935
AB  - We prove a formula for Reidemeister numbers of virtually polycyclic groups that generalises known formulas for finite groups and torsion-free virtually polycyclic groups. Based on this formula, we introduce a new invariant which we call the tamed Reidemeister number, and prove that its associated zeta functions are rational on virtually polycyclic groups. We also show that this implies rationality of (ordinary) Reidemeister zeta functions, as well as rationality of Nielsen zeta functions on compact infra-solvmanifolds.
ER  -