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PreprintCC BY 4.0arXiv

Presentations for sisters of one-cusped Picard modular groups

  • Jackson Young
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Abstract

We generalize a method devised by Mark and Paupert for finding group presentations of the single-cusped standard Picard modular groups PU(2, 1; π’ͺd), where π’ͺd is the ring of integers of β„š(i√d), to a more general class of lattices in PU(2, 1). As an application, we give presentations for the so-called \textit{sisters} of the standard Picard modular groups for d=3, 7, and $11$. In particular, this shows that the sister of PU(2, 1; π’ͺ7) is a perfect group.

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Cite

BibTeXRIS
BibTeX
@misc{young2026presentations,
  title = {{Presentations for sisters of one-cusped Picard modular groups}},
  author = {Jackson Young},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40568},
  eprint = {2610.40568},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40568},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Presentations for sisters of one-cusped Picard modular groups
AU  - Jackson Young
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.40568
UR  - https://arxiv.org/abs/2610.40568
AB  - We generalize a method devised by Mark and Paupert for finding group presentations of the single-cusped standard Picard modular groups PU(2, 1; π’ͺ_d), where π’ͺ_d is the ring of integers of β„š(i√d), to a more general class of lattices in PU(2, 1). As an application, we give presentations for the so-called \textit{sisters} of the standard Picard modular groups for d=3, 7, and $11$. In particular, this shows that the sister of PU(2, 1; π’ͺ_7) is a perfect group.
ER  -