Cycles in spherical Deligne complexes and application to $K(π,1)$-conjecture for Artin groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912190085201920 |
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| author | Huang, Jingyin |
| author_facet | Huang, Jingyin |
| contents | We introduce a method of finding large non-positively curved subcomplexes in certain spherical Deligne complexes, which is effective for studying fillings of certain 6-cycles in spherical Deligne complexes. As applications, we show the $K(π,1)$-conjecture holds for all 3-dimensional hyperbolic type Artin groups, except one single example; and the conjecture holds for all quasi-Lannér hyperbolic type Artin groups up to dimension 4. In higher dimension, we show the $K(π,1)$-conjecture for Artin groups whose Coxeter diagrams are complete bipartite (edge labels can be arbitrary), answering a question of J. McCammond. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_12068 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cycles in spherical Deligne complexes and application to $K(π,1)$-conjecture for Artin groups Huang, Jingyin Group Theory Geometric Topology Metric Geometry We introduce a method of finding large non-positively curved subcomplexes in certain spherical Deligne complexes, which is effective for studying fillings of certain 6-cycles in spherical Deligne complexes. As applications, we show the $K(π,1)$-conjecture holds for all 3-dimensional hyperbolic type Artin groups, except one single example; and the conjecture holds for all quasi-Lannér hyperbolic type Artin groups up to dimension 4. In higher dimension, we show the $K(π,1)$-conjecture for Artin groups whose Coxeter diagrams are complete bipartite (edge labels can be arbitrary), answering a question of J. McCammond. |
| title | Cycles in spherical Deligne complexes and application to $K(π,1)$-conjecture for Artin groups |
| topic | Group Theory Geometric Topology Metric Geometry |
| url | https://arxiv.org/abs/2405.12068 |