From Trees to Tripods: Proof of $K(π,1)$ for Artin groups with $ABI$-type spherical parabolics
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908843088281600 |
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| author | Hoda, Nima Huang, Jingyin |
| author_facet | Hoda, Nima Huang, Jingyin |
| contents | We reduce the $K(π,1)$-conjecture for all Artin groups with tree Coxeter diagrams to properties of Artin groups with tripod-shaped Coxeter diagrams. Combining this reduction theorem and properties of braid groups in previous works of Charney, Crisp-McCammond, Haettel and the second named author, we deduce that the $K(π,1)$-conjecture holds for every Artin group whose spherical parabolic subgroups avoid type $D_n$ ($n \ge 4$) and the exceptional types.
The reduction theorem relies on producing a ``tower'' of injective metric spaces from a single Artin group. The construction of such a tower relies on two ingredients of independent interests: a notion of combinatorial convexity and a Bestvina-type inequality, in certain injective orthoscheme complexes. These ingredients further rely on the use of structural properties of bi-Helly graphs (also known as absolute bipartite retracts) developed in joint work of the first named author with Munro. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_17983 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | From Trees to Tripods: Proof of $K(π,1)$ for Artin groups with $ABI$-type spherical parabolics Hoda, Nima Huang, Jingyin Group Theory Geometric Topology Metric Geometry We reduce the $K(π,1)$-conjecture for all Artin groups with tree Coxeter diagrams to properties of Artin groups with tripod-shaped Coxeter diagrams. Combining this reduction theorem and properties of braid groups in previous works of Charney, Crisp-McCammond, Haettel and the second named author, we deduce that the $K(π,1)$-conjecture holds for every Artin group whose spherical parabolic subgroups avoid type $D_n$ ($n \ge 4$) and the exceptional types. The reduction theorem relies on producing a ``tower'' of injective metric spaces from a single Artin group. The construction of such a tower relies on two ingredients of independent interests: a notion of combinatorial convexity and a Bestvina-type inequality, in certain injective orthoscheme complexes. These ingredients further rely on the use of structural properties of bi-Helly graphs (also known as absolute bipartite retracts) developed in joint work of the first named author with Munro. |
| title | From Trees to Tripods: Proof of $K(π,1)$ for Artin groups with $ABI$-type spherical parabolics |
| topic | Group Theory Geometric Topology Metric Geometry |
| url | https://arxiv.org/abs/2602.17983 |