On spherical Deligne complexes of type $D_n$
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910482659540992 |
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| author | Huang, Jingyin |
| author_facet | Huang, Jingyin |
| contents | Let $Δ$ be the Artin complex of the Artin group of type $D_n$. This complex is also called the spherical Deligne complex of type $D_n$. We show certain types of 6-cycles in the 1-skeleton of $Δ$ either have a center, which is a vertex adjacent to each vertex of the 6-cycle, or a quasi-center, which is a vertex adjacent to three of the alternating vertices of the 6-cycle. This will be a key ingredient in proving $K(π,1)$-conjecture for several classes of Artin groups in a companion article.
As a consequence, we also deduce that certain 2-dimensional relative Artin complex inside the $D_n$-type Artin complex, endowed with the induced Moussong metric, is CAT$(1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_11374 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On spherical Deligne complexes of type $D_n$ Huang, Jingyin Group Theory Geometric Topology Let $Δ$ be the Artin complex of the Artin group of type $D_n$. This complex is also called the spherical Deligne complex of type $D_n$. We show certain types of 6-cycles in the 1-skeleton of $Δ$ either have a center, which is a vertex adjacent to each vertex of the 6-cycle, or a quasi-center, which is a vertex adjacent to three of the alternating vertices of the 6-cycle. This will be a key ingredient in proving $K(π,1)$-conjecture for several classes of Artin groups in a companion article. As a consequence, we also deduce that certain 2-dimensional relative Artin complex inside the $D_n$-type Artin complex, endowed with the induced Moussong metric, is CAT$(1)$. |
| title | On spherical Deligne complexes of type $D_n$ |
| topic | Group Theory Geometric Topology |
| url | https://arxiv.org/abs/2405.11374 |