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PreprintCC BY 4.0arXiv

Reversibility and its asymptotic counting in Picard group

  • Debattam Das
  • Krishnendu Gongopadhyay
  • Anirban Mukhopadhyay
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Abstract

We investigate reversible elements in the Picard modular group PSL(2,ℤ[i]). We show that reversibility coincides with strong reversibility for Kleinian groups, in particular for the Picard group. We classify reversible elements in the Picard group and characterize loxodromic reversible elements up to conjugacy. We prove that each such conjugacy class contains exactly eight special representatives. We also obtain asymptotic estimates for the number of reversible conjugacy classes with bounded trace.

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Cite

BibTeXRIS
BibTeX
@misc{das2026reversibility,
  title = {{Reversibility and its asymptotic counting in Picard group}},
  author = {Debattam Das and Krishnendu Gongopadhyay and Anirban Mukhopadhyay},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11398},
  eprint = {2609.11398},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11398},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Reversibility and its asymptotic counting in Picard group
AU  - Debattam Das
AU  - Krishnendu Gongopadhyay
AU  - Anirban Mukhopadhyay
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.11398
UR  - https://arxiv.org/abs/2609.11398
AB  - We investigate reversible elements in the Picard modular group PSL(2,ℤ[i]). We show that reversibility coincides with strong reversibility for Kleinian groups, in particular for the Picard group. We classify reversible elements in the Picard group and characterize loxodromic reversible elements up to conjugacy. We prove that each such conjugacy class contains exactly eight special representatives. We also obtain asymptotic estimates for the number of reversible conjugacy classes with bounded trace.
ER  -