Classifying spaces for families of abelian subgroups of braid groups, RAAGs and graphs of abelian groups
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912125058809856 |
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| author | Álvarez, Porfirio L. León |
| author_facet | Álvarez, Porfirio L. León |
| contents | Given a group $G$ and an integer $n\geq 0$ we consider the family $\mathcal{F}_n$ of all virtually abelian subgroups of $G$ of rank at most $n$. In this article we prove that for each $n\ge2$ the Bredon cohomology, with respect to the family $\mathcal{F}_n$, of a free abelian group with rank $k > n$ is nontrivial in dimension $k+n$; this answers a question of Corob Cook, Moreno, Nucinkis and Pasini. As an application, we compute the minimal dimension of a classifying space for the family $\mathcal{F}_n$ for braid groups, right-angled Artin groups, and graphs of groups whose vertex groups are infinite finitely generated virtually abelian groups, for all $n\ge2$. The main tools that we use are the Mayer-Vietoris sequence for Bredon cohomology, Bass-Serre theory, and the Lück-Weiermann construction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_14315 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Classifying spaces for families of abelian subgroups of braid groups, RAAGs and graphs of abelian groups Álvarez, Porfirio L. León Group Theory Algebraic Topology Given a group $G$ and an integer $n\geq 0$ we consider the family $\mathcal{F}_n$ of all virtually abelian subgroups of $G$ of rank at most $n$. In this article we prove that for each $n\ge2$ the Bredon cohomology, with respect to the family $\mathcal{F}_n$, of a free abelian group with rank $k > n$ is nontrivial in dimension $k+n$; this answers a question of Corob Cook, Moreno, Nucinkis and Pasini. As an application, we compute the minimal dimension of a classifying space for the family $\mathcal{F}_n$ for braid groups, right-angled Artin groups, and graphs of groups whose vertex groups are infinite finitely generated virtually abelian groups, for all $n\ge2$. The main tools that we use are the Mayer-Vietoris sequence for Bredon cohomology, Bass-Serre theory, and the Lück-Weiermann construction. |
| title | Classifying spaces for families of abelian subgroups of braid groups, RAAGs and graphs of abelian groups |
| topic | Group Theory Algebraic Topology |
| url | https://arxiv.org/abs/2304.14315 |