Resolutions of orbifold generalised triangle groups

Abstract

Let T=⟨ f,g| f^l=g^m=Wⁿ=1⟩ be a generalised triangle group in which W is the Farey word of a rational slope q=r/s, and suppose that T is the fundamental group of a hyperbolic $3$-orbifold whose singular locus has no spherical vertices. Let G=⟨ f,g| f^l=g^m=Wⁿ⟩ be the associated central extension. We compute an explicit free resolution of the trivial module over ℤG of minimal length, with differentials given in closed form by Fox derivatives. From this resolution we deduce an explicit free resolution over ℤT which is $2$-periodic in degrees at least two, and we compute the integral homology and cohomology of both groups. Strebel treated the classical family q=0/1 in 1983. Since then only the family q=1/2 has been treated, by Ellis and Williams, and for that family our resolution over ℤT has strictly smaller rank in every degree.

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BibTeX RIS

BibTeX

@misc{carroll2026resolutions,
  title = {{Resolutions of orbifold generalised triangle groups}},
  author = {Sean P. Carroll and Thomas Csizmadia and Jeroen Schillewaert},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.20932},
  eprint = {2610.20932},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.20932},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Resolutions of orbifold generalised triangle groups
AU  - Sean P. Carroll
AU  - Thomas Csizmadia
AU  - Jeroen Schillewaert
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.20932
UR  - https://arxiv.org/abs/2610.20932
AB  - Let T=⟨ f,g| f^l=g^m=W^n=1⟩ be a generalised triangle group in which W is the Farey word of a rational slope q=r/s, and suppose that T is the fundamental group of a hyperbolic $3$-orbifold whose singular locus has no spherical vertices. Let G=⟨ f,g| f^l=g^m=W^n⟩ be the associated central extension. We compute an explicit free resolution of the trivial module over ℤG of minimal length, with differentials given in closed form by Fox derivatives. From this resolution we deduce an explicit free resolution over ℤT which is $2$-periodic in degrees at least two, and we compute the integral homology and cohomology of both groups. Strebel treated the classical family q=0/1 in 1983. Since then only the family q=1/2 has been treated, by Ellis and Williams, and for that family our resolution over ℤT has strictly smaller rank in every degree.
ER  -

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