The automorphism group of a free-by-cyclic groups in rank 2

Abstract

Let $\varphi$ be an automorphism of a free group $F_n$ of rank $n$, and let $M_{\varphi }=F_n{⋊}_{\varphi }\mathbb{Z}$ be the corresponding mapping torus of $\varphi$. We study the group $Out(M_{\varphi })$ under certain technical conditions on $\varphi$. Moreover, in the case of rank 2, we classify the cases when this group is finite or virtually cyclic, depending on the conjugacy class of the image of $\varphi$ in $G{L}_2(\mathbb{Z})$.

14 pages