Higher incoherence of the automorphism groups of a free group
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866908512816201728 |
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| author | Vidussi, Stefano |
| author_facet | Vidussi, Stefano |
| contents | Let $F_n$ be the free group on $n \geq 2$ generators. We show that for all $1 \leq m \leq 2n-3$ (respectively for all $1 \leq m \leq 2n-4$) there exists a subgroup of $\operatorname{Aut}(F_n)$ (respectively $\operatorname{Out}(F_n)$) which has finiteness of type $F_{m}$ but not of type $FP_{m+1}(\mathbb{Q})$, hence it is not $m$-coherent. In both cases, the new result is the upper bound $m= 2n-3$ (respectively $m = 2n-4$), as it cannot be obtained by embedding direct products of free noncyclic groups, and certifies higher incoherence up to the virtual cohomological dimension and is therefore sharp. As a tool of the proof, we discuss the existence and nature of multiple inequivalent extensions of a suitable finite-index subgroup $K_4$ of $\operatorname{Aut}(F_2)$ (isomorphic to the quotient of the pure braid group on four strands by its center): the fiber of four of these extensions arise from the strand-forgetting maps on the braid groups, while a fifth is related with the Cardano-Ferrari epimorphism. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_05118 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Higher incoherence of the automorphism groups of a free group Vidussi, Stefano Group Theory Geometric Topology Let $F_n$ be the free group on $n \geq 2$ generators. We show that for all $1 \leq m \leq 2n-3$ (respectively for all $1 \leq m \leq 2n-4$) there exists a subgroup of $\operatorname{Aut}(F_n)$ (respectively $\operatorname{Out}(F_n)$) which has finiteness of type $F_{m}$ but not of type $FP_{m+1}(\mathbb{Q})$, hence it is not $m$-coherent. In both cases, the new result is the upper bound $m= 2n-3$ (respectively $m = 2n-4$), as it cannot be obtained by embedding direct products of free noncyclic groups, and certifies higher incoherence up to the virtual cohomological dimension and is therefore sharp. As a tool of the proof, we discuss the existence and nature of multiple inequivalent extensions of a suitable finite-index subgroup $K_4$ of $\operatorname{Aut}(F_2)$ (isomorphic to the quotient of the pure braid group on four strands by its center): the fiber of four of these extensions arise from the strand-forgetting maps on the braid groups, while a fifth is related with the Cardano-Ferrari epimorphism. |
| title | Higher incoherence of the automorphism groups of a free group |
| topic | Group Theory Geometric Topology |
| url | https://arxiv.org/abs/2311.05118 |