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PreprintCC BY-NC-ND 4.0HAL

A cube complex for virtual Artin groups

  • JosΓ© Galvez Mateos
  • Luis Paris
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Abstract

Let Ξ“ be a Coxeter graph, and let S be its vertex set. An elemental complex over Ξ“ is an abstract simplicial complex π’Ÿ on S that contains every pair {s, t} with s β‰  t and ms,t β‰  ∞. To each elemental complex π’Ÿ, we associate a cube complex Ξžπ’Ÿ on which the virtual Artin group VA [Ξ“] acts. We prove that Ξžπ’Ÿ is connected, simply connected, and locally regular; that the action of VA[Ξ“] on Ξžπ’Ÿ is regular, cocompact, and by isometries; and that the stabilizers of cells are parabolic subgroups. We finally prove that Ξžπ’Ÿ is CAT(0) if and only if π’Ÿ is a flag complex.

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Cite

BibTeXRIS
BibTeX
@misc{mateos2026cube,
  title = {{A cube complex for virtual Artin groups}},
  author = {JosΓ© Galvez Mateos and Luis Paris},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.24840},
  eprint = {2607.24840},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05703062v3},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - A cube complex for virtual Artin groups
AU  - JosΓ© Galvez Mateos
AU  - Luis Paris
PY  - 2026
DA  - 2026/09/21
PB  - HAL
DO  - 10.48550/arxiv.2607.24840
UR  - https://hal.science/hal-05703062v3
AB  - Let Ξ“ be a Coxeter graph, and let S be its vertex set. An elemental complex over Ξ“ is an abstract simplicial complex π’Ÿ on S that contains every pair {s, t} with s β‰  t and m_s,t β‰  ∞. To each elemental complex π’Ÿ, we associate a cube complex Ξ_π’Ÿ on which the virtual Artin group VA [Ξ“] acts. We prove that Ξ_π’Ÿ is connected, simply connected, and locally regular; that the action of VA[Ξ“] on Ξ_π’Ÿ is regular, cocompact, and by isometries; and that the stabilizers of cells are parabolic subgroups. We finally prove that Ξ_π’Ÿ is CAT(0) if and only if π’Ÿ is a flag complex.
ER  -