Higher incoherence of the automorphism groups of a free group
| Autor: | Vidussi, Stefano |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 suitable finite-index subgroups of $\operatorname{Aut}(F_2)$. Comment: The results on virtual free-by-free structures of $\operatorname{Aut}(F_2)$, and their relation with properties of the braid group $B_4$, have been sharpened. Various typos and inaccuracies have been corrected |
| Databáze: | arXiv |
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