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| Format: | Preprint |
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2021
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| Online Access: | https://arxiv.org/abs/2109.07891 |
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| _version_ | 1866915067286519808 |
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| author | Haettel, Thomas |
| author_facet | Haettel, Thomas |
| contents | Starting with a lattice with an action of $\mathbb{Z}$ or $\mathbb{R}$, we build a Helly graph or an injective metric space. We deduce that the $\ell^\infty$ orthoscheme complex of any bounded graded lattice is injective. We also prove a Cartan-Hadamard result for locally injective metric spaces. We apply this to show that any Garside group or any FC type Artin group acts on an injective metric space and on a Helly graph. We also deduce that the natural piecewise $\ell^\infty$ metric on any Euclidean building of type $\tilde{A_n}$ extended, $\tilde{B_n}$, $\tilde{C_n}$ or $\tilde{D_n}$ is injective, and its thickening is a Helly graph. Concerning Artin groups of Euclidean types $\tilde{A_n}$ and $\tilde{C_n}$, we show that the natural piecewise $\ell^\infty$ metric on the Deligne complex is injective, the thickening is a Helly graph, and it admits a convex bicombing. This gives a metric proof of the $K(π,1)$ conjecture, as well as several other consequences usually known when the Deligne complex has a CAT(0) metric. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_07891 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Lattices, injective metrics and the $K(π,1)$ conjecture Haettel, Thomas Group Theory Geometric Topology Metric Geometry 52A35, 20E42, 05B35, 06A12, 20F65 Starting with a lattice with an action of $\mathbb{Z}$ or $\mathbb{R}$, we build a Helly graph or an injective metric space. We deduce that the $\ell^\infty$ orthoscheme complex of any bounded graded lattice is injective. We also prove a Cartan-Hadamard result for locally injective metric spaces. We apply this to show that any Garside group or any FC type Artin group acts on an injective metric space and on a Helly graph. We also deduce that the natural piecewise $\ell^\infty$ metric on any Euclidean building of type $\tilde{A_n}$ extended, $\tilde{B_n}$, $\tilde{C_n}$ or $\tilde{D_n}$ is injective, and its thickening is a Helly graph. Concerning Artin groups of Euclidean types $\tilde{A_n}$ and $\tilde{C_n}$, we show that the natural piecewise $\ell^\infty$ metric on the Deligne complex is injective, the thickening is a Helly graph, and it admits a convex bicombing. This gives a metric proof of the $K(π,1)$ conjecture, as well as several other consequences usually known when the Deligne complex has a CAT(0) metric. |
| title | Lattices, injective metrics and the $K(π,1)$ conjecture |
| topic | Group Theory Geometric Topology Metric Geometry 52A35, 20E42, 05B35, 06A12, 20F65 |
| url | https://arxiv.org/abs/2109.07891 |