Parabolic subgroups and word problem in virtual Artin groups
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911473058447360 |
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| author | Mateos, José Gálvez Gavazzi, Federica Paris, Luis |
| author_facet | Mateos, José Gálvez Gavazzi, Federica Paris, Luis |
| contents | We begin by establishing two fundamental results on standard parabolic subgroups of virtual Artin groups. We first show that a standard parabolic subgroup is naturally isomorphic to a virtual Artin group. Second, we prove that the intersection of two standard parabolic subgroups is a standard parabolic subgroup. Our main result is that, if all free of infinity standard parabolic subgroups of a given virtual Artin group VA[Γ] have a solvable word problem, then VA[Γ] itself has a solvable word problem. It follows that virtual Artin groups of FC type and, more generally, of affine-FC type, have a solvable word problem. We also prove that, if a virtual Artin group VA[Γ] has a solvable word problem, then the strong membership problem for any standard parabolic subgroup in VA[Γ] is solvable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_23819 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Parabolic subgroups and word problem in virtual Artin groups Mateos, José Gálvez Gavazzi, Federica Paris, Luis Group Theory We begin by establishing two fundamental results on standard parabolic subgroups of virtual Artin groups. We first show that a standard parabolic subgroup is naturally isomorphic to a virtual Artin group. Second, we prove that the intersection of two standard parabolic subgroups is a standard parabolic subgroup. Our main result is that, if all free of infinity standard parabolic subgroups of a given virtual Artin group VA[Γ] have a solvable word problem, then VA[Γ] itself has a solvable word problem. It follows that virtual Artin groups of FC type and, more generally, of affine-FC type, have a solvable word problem. We also prove that, if a virtual Artin group VA[Γ] has a solvable word problem, then the strong membership problem for any standard parabolic subgroup in VA[Γ] is solvable. |
| title | Parabolic subgroups and word problem in virtual Artin groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2602.23819 |