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PreprintCC BY 4.0arXiv

On the type of a complex hyperbolic triangle group

  • Wei Liao
  • John R Parker
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Abstract

In this paper we consider groups of complex hyperbolic isometries generated by three complex involutions. Our first main result states that if the group is discrete and at least two of the mirrors of complex reflections intersect, then the group may be put into a normal form involving the ordering of traces of short, even length words. Our second main result refines a conjecture of Schwartz (proved by Grossi) that determines the type of the group in terms of the angles between the mirrors. That is, in the parameter space of complex triangles with given angles, it determines which of two given words becomes elliptic first.

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Cite

BibTeXRIS
BibTeX
@misc{liao2026type,
  title = {{On the type of a complex hyperbolic triangle group}},
  author = {Wei Liao and John R Parker},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.03422},
  eprint = {2610.03422},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.03422},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the type of a complex hyperbolic triangle group
AU  - Wei Liao
AU  - John R Parker
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2610.03422
UR  - https://arxiv.org/abs/2610.03422
AB  - In this paper we consider groups of complex hyperbolic isometries generated by three complex involutions. Our first main result states that if the group is discrete and at least two of the mirrors of complex reflections intersect, then the group may be put into a normal form involving the ordering of traces of short, even length words. Our second main result refines a conjecture of Schwartz (proved by Grossi) that determines the type of the group in terms of the angles between the mirrors. That is, in the parameter space of complex triangles with given angles, it determines which of two given words becomes elliptic first.
ER  -