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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2410.16390 |
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| _version_ | 1866912081184292864 |
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| author | Arenas, Macarena |
| author_facet | Arenas, Macarena |
| contents | We show that, under suitable hypotheses, the coned-off spaces associated to $C(9)$ cubical small-cancellation presentations are aspherical, and use this to provide classifying spaces, or classifying spaces for proper actions, for their fundamental groups. Along the way, we show that the Cohen--Lyndon property holds for the subgroups of the fundamental group of a non-positively curved cube complex associated to a $C(9)$ cubical presentation, and thus obtain near-sharp upper and lower bounds for the (rational) cohomological dimension of these quotients.
We apply these results to give an alternative construction of compact $K(π,1)$ for Artin groups with no labels in $\{3,4\}$, from which a new direct sum decomposition for their homology and cohomology with various coefficients above a certain dimension follows. We also address a question of Wise about the virtual torsion-freeness of cubical small-cancellation groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_16390 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asphericity of cubical presentations: the general case Arenas, Macarena Group Theory Geometric Topology 20F06, 20F67 We show that, under suitable hypotheses, the coned-off spaces associated to $C(9)$ cubical small-cancellation presentations are aspherical, and use this to provide classifying spaces, or classifying spaces for proper actions, for their fundamental groups. Along the way, we show that the Cohen--Lyndon property holds for the subgroups of the fundamental group of a non-positively curved cube complex associated to a $C(9)$ cubical presentation, and thus obtain near-sharp upper and lower bounds for the (rational) cohomological dimension of these quotients. We apply these results to give an alternative construction of compact $K(π,1)$ for Artin groups with no labels in $\{3,4\}$, from which a new direct sum decomposition for their homology and cohomology with various coefficients above a certain dimension follows. We also address a question of Wise about the virtual torsion-freeness of cubical small-cancellation groups. |
| title | Asphericity of cubical presentations: the general case |
| topic | Group Theory Geometric Topology 20F06, 20F67 |
| url | https://arxiv.org/abs/2410.16390 |