Finiteness properties of centralisers in $\mathrm{Out}(F_N)$ and automorphisms of free-by-cyclic groups

Abstract

Let $F$ be a finitely generated free group. We prove that the centraliser of any element of $\mathrm{Out}(F)$ or $\mathrm{Aut}(F)$ is of type $\rm{VF}$. We also prove that the (outer) automorphism group of any free-by-cyclic group $M=F\rtimes_φ\mathbb{Z}$ is of type $\rm{VF}$, and so are the McCool subgroups of $\mathrm{Out}(M)$. This relies on the construction of a new $\mathrm{Out}(M)$-invariant splitting of $M$, that enables us to reduce the proof of the above statements to the case where the outer class of $φ$ has finite order.