Higher incoherence of the automorphism groups of a free group

Abstract 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 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.