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PreprintCC BY-NC-ND 4.0arXiv

The Conjugacy Problem for Recursively Presented Virtually Soluble Groups of Finite Prüfer Rank

  • M. Brescia
  • E. Ingrosso
  • M. Trombetti
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Abstract

We solve a long-standing conjugacy problem for finitely presented soluble groups of finite Prüfer rank, recorded as Problem~9.52 in the Kourovka Notebook, in the stronger setting of recursively presented finitely generated virtually soluble groups. Robinson proved that, in every recursively presented finitely generated virtually soluble minimax group, conjugacy to any fixed element is decidable, while explicitly lefting open whether the procedure could be made uniform in the target element. We establish this uniformity. More specifically, we prove that every recursively presented finitely generated virtually soluble group of finite Prüfer rank has decidable Conjugacy Problem.

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Cite

BibTeXRIS
BibTeX
@misc{brescia2026conjugacy,
  title = {{The Conjugacy Problem for Recursively Presented Virtually Soluble Groups of Finite Prüfer Rank}},
  author = {M. Brescia and E. Ingrosso and M. Trombetti},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29563},
  eprint = {2610.29563},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29563},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - The Conjugacy Problem for Recursively Presented Virtually Soluble Groups of Finite Prüfer Rank
AU  - M. Brescia
AU  - E. Ingrosso
AU  - M. Trombetti
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29563
UR  - https://arxiv.org/abs/2610.29563
AB  - We solve a long-standing conjugacy problem for finitely presented soluble groups of finite Prüfer rank, recorded as Problem~9.52 in the Kourovka Notebook, in the stronger setting of recursively presented finitely generated virtually soluble groups. Robinson proved that, in every recursively presented finitely generated virtually soluble minimax group, conjugacy to any fixed element is decidable, while explicitly lefting open whether the procedure could be made uniform in the target element. We establish this uniformity. More specifically, we prove that every recursively presented finitely generated virtually soluble group of finite Prüfer rank has decidable Conjugacy Problem.
ER  -