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Higher incoherence of the automorphism groups of a free group
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 01 September 2025, pp. 1-28
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- Article
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Let Fn 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 Fm 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 K4 of 
${\operatorname{Aut}(F_2)}$ (isomorphic to the quotient of the pure braid group on four strands by its centre): the fibre 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.
