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 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.